Dynamic Games with Almost Perfect Information

Wei He∗ Yeneng Sun†

This version: October 8, 2015

Job Market Paper

Abstract

This paper aims to solve two fundamental problems on finite or infinite horizon

dynamic games with complete information. Under some mild conditions, we

prove (1) the existence of subgame-perfect equilibria in general dynamic games

with simultaneous moves (i.e. almost perfect information), and (2) the existence

of pure-strategy subgame-perfect equilibria in perfect-information dynamic games

with uncertainty. Our results go beyond previous works on continuous dynamic

games in the sense that public randomization and the continuity requirement on the

state variables are not needed. As an illustrative application, a dynamic stochastic

oligopoly market with intertemporally dependent payoffs is considered.

∗Department of Economics, The University of Iowa, W249 Pappajohn Business Building, Iowa City,

IA 52242. E-mail: he.wei2126@gmail.com.†Department of Economics, National University of Singapore, 1 Arts Link, Singapore 117570. Email:

ynsun@nus.edu.sg

1

Contents

1 Introduction 3

2 Model and main result 6

3 Dynamic oligopoly market with sticky prices 9

4 Variations of the main result 10

4.1 Dynamic games with partially perfect information and a generalized

ARM condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Continuous dynamic games with partially perfect information . . . 13

5 Appendix 15

5.1 Technical preparations . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2 Discontinuous games with endogenous stochastic sharing rules . . . 33

5.3 Proofs of Theorem 1 and Proposition 1 . . . . . . . . . . . . . . . . 37

5.3.1 Backward induction . . . . . . . . . . . . . . . . . . . . . . 37

5.3.2 Forward induction . . . . . . . . . . . . . . . . . . . . . . . 37

5.3.3 Infinite horizon case . . . . . . . . . . . . . . . . . . . . . . 42

5.4 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.5 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . 61

References 63

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1 Introduction

Dynamic games with complete information, where players observe the history and

move simultaneously or alternatively in finite or infinite horizon, arise naturally in

many situations. As noted in Fudenberg and Tirole (1991, Chapter 4) and Harris,

Reny and Robson (1995), these games form a general class of regular dynamic

games with many applications in economics, political science, and biology. The

associated notion of subgame-perfect equilibrium is a fundamental game-theoretic

concept. For games with finitely many actions and stages, Selten (1965) showed

the existence of subgame-perfect equilibria. The infinite horizon but finite-action

case was covered by Fudenberg and Levine (1983).

Since the agents in many economic models need to make continuous choices,

it is important to consider dynamic games with general action spaces. However,

Harris, Reny and Robson (1995) presented a simple example without subgame-

perfect equilibrium, where the game has two players in each of the two stages with

only one player having a continuous choice set.1 Thus, the existence of subgame-

perfect equilibria under some suitable conditions remains an open problem even

for two-stage dynamic games. The purpose of this paper is to prove the existence

of subgame-perfect equilibria for general dynamic games with simultaneous or

alternative moves in finite or infinite horizon under some suitable conditions.

We shall adopt the general forms of intertemporal preferences, requiring neither

stationarity nor additive separability.

For deterministic games with perfect information (i.e., the players move

alternatively), the existence of pure-strategy subgame-perfect equilibria was shown

in Harris (1985), Hellwig and Leininger (1987), Borgers (1989, 1991) and Hellwig et

al. (1990) with the model parameters being continuous in actions. However, if the

“deterministic” assumption is dropped by introducing a passive player - Nature,

then pure-strategy subgame-perfect equilibrium need not exist as shown by a four-

stage game in Harris, Reny and Robson (1995, p. 538). In fact, Luttmer and

Mariotti (2003) even demonstrated the nonexistence of mixed-strategy subgame-

perfect equilibrium in a five-stage game.2 Thus, it is still an open problem to show

the existence of (pure or mixed-strategy) subgame-perfect equilibria in (finite or

infinite horizon) perfect-information dynamic games with uncertainty under some

general conditions.

1As noted in Section 2 of Harris, Reny and Robson (1995), such a dynamic game with continuouschoices provides a minimal nontrivial counterexample; see also Exercise 13.4 in Fudenberg and Tirole(1991) for another counterexample of Harris.

2All those counterexamples show that various issues arise when one considers general dynamic games.In the setting with incomplete information, even the equilibrium notion needs to be carefully treated;see Myerson and Reny (2015).

3

Harris, Reny and Robson (1995) considered continuous dynamic games with

almost perfect information in the sense that the players move simultaneously.

In such games, there are a finite number of active players and a passive player

(Nature), and all the relevant model parameters are assumed to be continuous in

both action and state variables (i.e., Nature’s moves). Harris, Reny and Robson

(1995) showed the existence of subgame-perfect correlated equilibria by introducing

a public randomization device.3 As mentioned above, they also demonstrated the

possible nonexistence of subgame-perfect equilibrium through a simple example.

This paper resolves the above two open problems (in both finite and infinite

horizon) not just for the class of continuous dynamic games as considered in

the earlier works, but more importantly, for general dynamic games in which

the relevant model parameters are assumed to be continuous in actions and

measurable in states.4 In particular, we show the existence of a subgame-

perfect equilibrium in a general dynamic game with almost perfect information

under some suitable conditions on the state transitions. Theorem 1 (and also

Proposition 2) below goes beyond earlier works on continuous dynamic games

by dropping public randomization and the continuity requirement on the state

variables. Thus, the class of games considered here includes general stochastic

games, where the stage payoffs are usually assumed to be continuous in actions and

measurable in states.5 Our Proposition 1 in Section 2 also presents some regularity

properties of the equilibrium payoff correspondences, including compactness and

upper hemicontinuity in the action variables.6 As an illustrative application of

Theorem 1, we consider a dynamic oligopoly market in which firms face stochastic

demand/cost and intertemporally dependent payoffs.

We work with the condition that the state transition in each period (except

for those periods with one active player) has a component with a suitable density

function with respect to some atomless reference measure. This condition is also

minimal in the particular sense that the existence result may fail to hold if (1) the

passive player, Nature, is not present in the model as shown in Harris, Reny and

3See also Mariotti (2000) and Reny and Robson (2002).4Since the agents need to make optimal choices, the continuity assumption in terms of actions

is natural and widely adopted. However, the state variable is not a choice variable, and thus itis unnecessary to impose the state continuity requirement in a general model. Note that the statemeasurability assumption is the minimal regularity condition one would expect for the model parameters.In addition, we point out that the proof of our main result in the case with state continuity as inSubsection 5.5 is much simpler than that of the general case.

5Proposition 2 below does imply a new existence result on subgame-perfect equilibria for a generalstochastic game; see Remark 2 in Subsection 4.1.

6Such an upper hemicontinuity property in terms of correspondences of equilibrium payoffs, oroutcomes, or correlated strategies has been the key for proving the relevant existence results as inHarris (1985), Hellwig and Leininger (1987), Borgers (1989, 1991), Hellwig et al. (1990), Harris, Renyand Robson (1995) and Mariotti (2000).

4

Robson (1995), or (2) with the presence of Nature, the reference measure is not

atomless as shown in Luttmer and Mariotti (2003).

For the special class of continuous dynamic games with almost perfect

information, we can weaken the atomless reference measure condition slightly.

In particular, we simply assume the state transition in each period (except for

those periods with one active player) to be an atomless probability measure

for any given history, without the requirement of a common reference measure.

Note that our model allows the state history to fully influence all the model

parameters. Clearly, it covers the case with public randomization in the sense that

the state transition has an additional atomless component that is independently

and identically distributed across time, and does not enter the payoffs, state

transitions and action correspondences. Thus, we obtain the existence result in

Harris, Reny and Robson (1995) as a special case.

For dynamic games with almost perfect information, our main result allows

the players to take mixed strategies. However, for the special class of dynamic

games with perfect information,7 we obtain the existence of pure-strategy subgame-

perfect equilibria in Corollaries 2 and 3. When Nature is present, the only known

general existence result for dynamic games with perfect information is, to the

best of our knowledge, for continuous games with public randomization. On the

contrary, our Corollary 2 needs neither continuity in the state variables nor public

randomization. Furthermore, our Corollary 3 provides a new existence result for

continuous dynamic games with perfect information, which generalizes the results

of Harris (1985), Hellwig and Leininger (1987), Borgers (1989), and Hellwig et al.

(1990) to the case when Nature is present.

To prove Theorem 1, we follow the standard three-step procedure in obtaining

subgame-perfect equilibria of dynamic games, namely, backward induction, forward

induction, and approximation of infinite horizon by finite horizon. Because we

drop public randomization and the continuity requirement on the state variables,

new technical difficulties arise in each step of the proof. In the step of backward

induction, we obtain a new existence result for discontinuous games with stochastic

endogenous sharing rules, which extends the main result of Simon and Zame

(1990) by allowing the payoff correspondence to be measurable (instead of upper

hemicontinuous) in states. For forward induction, we need to obtain strategies

that are jointly measurable in history. When there is a public randomization

device, the joint measurability follows from the measurable version of Skorokhod’s

representation theorem and implicit function theorem respectively as in Harris,

7Dynamic games with perfect information do have wide applications. For some examples, see Phelpsand Pollak (1968) for an intergenerational bequest game, and Peleg and Yaari (1973) and Goldman(1980) for intrapersonal games in which consumers have changing preferences.

5

Reny and Robson (1995) and Reny and Robson (2002). Here we need to work with

the deep “measurable” measurable choice theorem of Mertens (2003). Our main

existence result for the finite-horizon case then follows immediately from backward

and forward inductions. Lastly, in order to extend the result to the infinite horizon

setting, we need to handle various subtle measurability issues due to the lack of

continuity in the state variables in our model, which is the most difficult part of

the proof for Theorem 1.8 As noted in Subsection 5.5 below, the proof for the

existence result in the case of continuous dynamic games (i.e., Proposition 3) is

much simpler than that of Theorem 1.

The rest of the paper is organized as follows. The model and main result are

presented in Section 2. An illustrative application of Theorem 1 to a dynamic

oligopoly market with stochastic demand/cost and intertemporally dependent

payoffs is given in Section 3. Section 4 provides several variations of the main

result. All the proofs are left in the Appendix.

2 Model and main result

In this section, we shall present the model for an infinite-horizon dynamic game

with almost perfect information.

The set of players is I0 = {0, 1, . . . , n}, where the players in I = {1, . . . , n} are

active and player 0 is Nature. All players move simultaneously. Time is discrete,

and indexed by t = 0, 1, 2, . . ..

The set of starting points is a closed set H0 = X0×S0, where X0 is a compact

metric space and S0 is a Polish space (that is, a complete separable metric space).9

At stage t ≥ 1, player i’s action will be chosen from a subset of a Polish space Xti

for each i ∈ I, and Xt =∏i∈I Xti. Nature’s action is chosen from a Polish space

St. Let Xt =∏

0≤k≤tXk and St =∏

0≤k≤t Sk. The Borel σ-algebras on Xt and

St are denoted by B(Xt) and B(St), respectively. Given t ≥ 0, a history up to the

stage t is a vector

ht = (x0, s0, x1, s1, . . . , xt, st) ∈ Xt × St.

The set of all such possible histories is denoted by Ht. For any t ≥ 0, Ht ⊆ Xt×St.For any t ≥ 1 and i ∈ I, let Ati be a measurable, nonempty and compact

valued correspondence from Ht−1 to Xti such that (1) Ati is sectionally continuous

8Because our relevant model parameters are only measurable in the state variables, the usual methodof approximating a limit continuous dynamic game by a sequence of finite games, as used in Hellwig etal. (1990), Borgers (1991) and Harris, Reny and Robson (1995), is not applicable in our setting.

9In each stage t ≥ 1, there will be a set of action profiles Xt and a set of states St. Without loss ofgenerality, we assume that the set of initial points is also a product space for notational consistency.

6

on Xt−1,10 and (2) Ati(ht−1) is the set of available actions for player i ∈ I given

the history ht−1.11 Let At =∏i∈I Ati. Then Ht = Gr(At) × St, where Gr(At) is

the graph of At.

For any x = (x0, x1, . . .) ∈ X∞, let xt = (x0, . . . , xt) ∈ Xt be the truncation of

x up to the period t. Truncations for s ∈ S∞ can be defined similarly. Let H∞ be

the subset of X∞ × S∞ such that (x, s) ∈ H∞ if (xt, st) ∈ Ht for any t ≥ 0. Then

H∞ is the set of all possible histories in the game.12

For any t ≥ 1, Nature’s action is given by a Borel measurable mapping ft0 from

Ht−1 to M(St) such that ft0 is sectionally continuous on Xt−1, where M(St) is

endowed with the topology induced by the weak convergence.13 For each t ≥ 0,

suppose that λt is a Borel probability measure on St and λt is atomless for t ≥ 1.

Let λt = ⊗0≤k≤tλt for t ≥ 0. We shall assume the following condition on the state

transitions.

Assumption 1 (Atomless Reference Measure (ARM)). A dynamic game is said

to satisfy the “atomless reference measure (ARM)” condition if for each t ≥ 1,

1. the probability ft0(·|ht−1) is absolutely continuous with respect to λt on St

with the Radon-Nikodym derivative ϕt0(ht−1, st) for all ht−1 ∈ Ht−1;14

2. the mapping ϕt0 is Borel measurable and sectionally continuous on Xt−1, and

integrably bounded in the sense that there is a λt-integrable function φt : St →R+ such that ϕt0(ht−1, st) ≤ φt(st) for any ht−1 ∈ Ht−1 and st ∈ St.

For each i ∈ I, the payoff function ui is a Borel measurable mapping from H∞

to R++ which is sectionally continuous on X∞ and bounded by γ > 0.15

When one considers a dynamic game with infinite horizon, the following

“continuity at infinity” condition is standard.16 In particular, all discounted

repeated games or stochastic games satisfy this condition.

10Suppose that Y1, Y2 and Y3 are all Polish spaces, and Z ⊆ Y1 × Y2. Denote Z(y1) = {y2 ∈Y2 : (y1, y2) ∈ Z} for any y1 ∈ Y1. A function (resp. correspondence) f : Z → Y3 is said to be sectionallycontinuous on Y2 if f(y1, ·) is continuous on Z(y1) for all y1 with Z(y1) 6= ∅. Similarly, one can definethe sectional upper hemicontinuity for a correspondence.

11Suppose that Y and Z are both Polish spaces, and Ψ is a correspondence from Y to Z. Hereafter,the measurability of Ψ, unless specifically indicated, is with respect to the Borel σ-algebra B(Y ) on Y .

12A finite horizon dynamic game can be regarded as a special case of an infinite horizon dynamicgame in the sense that the action correspondence Ati is point-valued for each player i ∈ I and t ≥ T forsome stage T ≥ 1; see, for example, Borgers (1989) and Harris, Reny and Robson (1995).

13For a Polish space A, M(A) denotes the set of all Borel probability measures on A, and 4(A) isthe set of all finite Borel measures on A.

14It is common to have a reference measure when one considers a game with uncountable states. Forexample, if St is a subset of Rl, then the Lebesgue measure is a natural reference measure.

15Since ui is bounded, we can assume that the value of the payoff function is strictly positive withoutloss of generality.

16See, for example, Fudenberg and Levine (1983).

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For any T ≥ 1, let

wT = supi∈I

(x,s)∈H∞(x,s)∈H∞xT−1=xT−1

sT−1=sT−1

|ui(x, s)− ui(x, s)|. (1)

Assumption 2 (Continuity at Infinity). A dynamic game is said to be “continuous

at infinity” if wT → 0 as T →∞.

For player i ∈ I, a strategy fi is a sequence {fti}t≥1 such that fti is a Borel

measurable mapping from Ht−1 to M(Xti) with fti(Ati(ht−1)|ht−1) = 1 for all

ht−1 ∈ Ht−1. A strategy profile f = {fi}i∈I is a combination of strategies of all

active players.

In any subgame, a strategy combination will generate a probability distribution

over the set of possible histories. This probability distribution is called the path

induced by the strategy combination in this subgame.

Definition 1. Suppose that a strategy profile f = {fi}i∈I and a history ht ∈ Ht are

given for some t ≥ 0. Let τt = δht, where δht is the probability measure concentrated

at the one point ht. If τt′ ∈M(Ht′) has already been defined for some t′ ≥ t, then

let

τt′+1 = τt′ � (⊗i∈I0f(t′+1)i).17

Finally, let τ ∈ M(H∞) be the unique probability measure on H∞ such that

MargHt′τ = τt′ for all t′ ≥ t. Then τ is called the path induced by f in the

subgame ht. For all i ∈ I,∫H∞

ui dτ is the payoff of player i in this subgame.

The notion of subgame-perfect equilibrium is defined as follows.

Definition 2 (SPE). A subgame-perfect equilibrium is a strategy profile f such

that for all i ∈ I, t ≥ 0, and λt-almost all ht ∈ Ht,18 player i cannot improve his

payoff in the subgame ht by a unilateral change in his strategy.19

17Denote ⊗i∈I0f(t′+1)i as a transition probability from Ht′ to M(Xt′+1). Notice that the strategyprofile is usually represented by a vector. For the notational simplicity later on, we assume that⊗i∈I0f(t′+1)i(·|ht′) represents the strategy profile in stage t′ + 1 for a given history ht′ ∈ Ht′ , where⊗i∈I0f(t′+1)i(·|ht′) is the product of the probability measures f(t′+1)i(·|ht′), i ∈ I0. If λ is a finitemeasure on X and ν is a transition probability from X to Y , then λ � ν is a measure on X × Y suchthat λ � ν(A×B) =

∫Aν(B|x)λ(dx) for any measurable subsets A ⊆ X and B ⊆ Y .

18A property is said to hold for λt-almost all ht = (xt, st) ∈ Ht if it is satisfied for λt-almost all st ∈ Stand all xt ∈ Ht(s

t).19When the state space is uncountable and has a reference measure, it is natural to consider the

optimality for almost all sub-histories in the probabilistic sense; see, for example, Abreu, Pearce andStacchetti (1990) and Footnote 4 therein.

8

The following theorem is our main result, which shows the existence of a

subgame-perfect equilibrium under the conditions of ARM and continuity at

infinity. Its proof is left in the appendix.

Theorem 1. If a dynamic game satisfies the ARM condition and is continuous at

infinity, then it possesses a subgame-perfect equilibrium.

Let Et(ht−1) be the set of subgame-perfect equilibrium payoffs in the subgame

ht−1. The following result demonstrates the compactness and upper hemicontinuity

properties of the correspondence Et.

Proposition 1. If a dynamic game satisfies the ARM condition and is continuous

at infinity, then Et is nonempty and compact valued, and essentially sectionally

upper hemicontinuous on Xt−1.20

3 Dynamic oligopoly market with sticky prices

In this section, we consider a dynamic oligopoly market in which firms face

stochastic demand/cost and intertemporally dependent payoffs. Such a model is

a variant of the well-known dynamic oligopoly models as considered in Green and

Porter (1984) and Rotemberg and Saloner (1986), which examined the response of

firms for demand fluctuations. The key feature of our example is the existence of

sticky price effect, which means that the desirability of the good from the demand

side could depend on the accumulated past output, and hence gives intertemporally

dependent payoff functions.

We consider a dynamic oligopoly market in which n firms produce a homoge-

neous good in an infinite-horizon setting. The inverse demand function is denoted

by Pt(Q1, . . . , Qt, st), whereQt is the industry output and st the observable demand

shock in period t. Notice that the price depends on the past outputs. One possible

reason could be that the desirability of consumers will be influenced by their

previous consumptions, and hence the price does not adjust instantaneously. We

assume that Pt is a bounded function which is continuous in (Q1, . . . , Qt) and

measurable in st. In period t, the shock st is selected from the set St = [at, bt].

We denote firm i’s output in period t by qti so that Qt =∑n

i=1 qti. The cost of

firm i in period t is cti(qti, st) given the output qti and the shock st, where cti is a

bounded function continuous in qti and measurable in st. The discount factor of

firm i is βi ∈ [0, 1).

20Suppose that Y1, Y2 and Y3 are all Polish spaces, and Z ⊆ Y1×Y2 and η is a Borel probability measureon Y1. Denote Z(y1) = {y2 ∈ Y2 : (y1, y2) ∈ Z} for any y1 ∈ Y1. A function (resp. correspondence)f : Z → Y3 is said to be essentially sectionally continuous on Y2 if f(y1, ·) is continuous on Z(y1) for η-almost all y1. Similarly, one can define the essential sectional upper hemicontinuity for a correspondence.

9

The timing of events is as follows.

1. At the beginning of period t, all firms learn the realization of st, which

is determined by the law of motion κt(·|s1, Q1, . . . , st−1, Qt−1). Suppose

that κt(·|s1, Q1, . . . , st−1, Qt−1) is absolutely continuous with respect to the

uniform distribution on St with density ϕt(s1, Q1, . . . , st−1, Qt−1, st), where

ϕt is bounded, continuous in (Q1, . . . , Qt−1) and measurable in (s1, . . . , st).

2. Firms then simultaneously choose the level of their output qt = (qt1, . . . , qtn),

where qti ∈ Ati(st, Qt−1) ⊆ Rl for i = 1, 2, . . . , n. In particular, the

correspondence Ati gives the available actions of firm i, which is nonempty

and compact valued, measurable in st, and continuous in Qt−1.

3. The strategic choices of all the firms then become common knowledge and

this one-period game is repeated.

In period t, given the shock st and the output {qk}1≤k≤t up to time t with

qk = (qk1, . . . , qkn), the payoff of firm i is

uti(q1, . . . , qt, st) =

Pt( n∑j=1

q1j , . . . ,

n∑j=1

qtj , st)− cti(qti, st)

qti.Given a sequence of outputs {qt}t≥1 and shocks {st}t≥1, firm i receives the payoff

u1i(q1, s1) +

∞∑t=2

βt−1i uti(q1, . . . , qt, st).

Remark 1. Our dynamic oligopoly model has a non-stationary structure. In

particular, the transitions and payoffs are history-dependent. The example

captures the scenario that the price of the homogeneous product does not adjust

instantaneously to the price indicated by its demand function at the given level

of output. For more applications with intertemporally dependent utilities, see, for

example, Ryder and Heal (1973), Fershtman and Kamien (1987) and Becker and

Murphy (1988). If the model is stationary and the inverse demand function only

depends the current output, then the example reduces to be the dynamic oligopoly

game with demand fluctuations as considered in Rotemberg and Saloner (1986).

By condition (1) above, the ARM condition is satisfied. It is also easy to see

that the game is continuous at infinity. By Theorem 1, we have the following result.

Corollary 1. The dynamic oligopoly market possesses a subgame-perfect equilib-

rium.

10

4 Variations of the main result

In this section, we will consider several variations of our main result.

In Subsection 4.1, we still consider dynamic games whose parameters are

continuous in actions and measurable in states. We partially relax the ARM

condition in two ways. First, we allow the possibility that there is only one

active player (but no Nature) at some stages, where the ARM type condition

is dropped. Second, we introduce an additional weakly continuous component on

the state transitions at any other stages. In addition, we allow the state transition

in each period to depend on the current actions as well as on the previous history.

Thus, we combine the models for dynamic games with perfect and almost perfect

information. We show the existence of a subgame perfect equilibrium such that

whenever there is only one active player at some stage, the player can play pure

strategy as part of the equilibrium strategies. As a byproduct, we obtain a new

existence result for stochastic games. The existence of pure-strategy subgame-

perfect equilibria for dynamic games with perfect information (with or without

Nature) is provided as an immediate corollary.

In Subsection 4.2, we consider the special case of continuous dynamic games

in the sense that all the model parameters are continuous in both action and

state variables. We can obtain the corresponding results under a slightly weaker

condition. All the previous existence results for continuous dynamic games with

perfect and almost perfect information are covered as our special cases.

We will follow the setting and notations in Section 2 as closely as possible. For

simplicity, we only describe the changes we need to make on the model. All the

proofs are left in the appendix.

4.1 Dynamic games with partially perfect information

and a generalized ARM condition

In this subsection, we will generalize the model in Section 2 in three directions.

The ARM condition is partially relaxed such that (1) perfect information may be

allowed in some stages, and (2) the state transitions have a weakly continuous

component in all other stages. In addition, the state transition in any period can

depend on the action profile in the current stage as well as on the previous history.

The fist change allows us to combine the models of dynamic games with perfect

and almost perfect information. The second generalization implies that the state

transitions need not be norm continuous on the Banach space of finite measures.

The last modification covers the model of stochastic games as a special case.

The changes are described below.

11

1. The state space is a product space of two Polish spaces; that is, St = St × Stfor each t ≥ 1.

2. For each i ∈ I, the action correspondence Ati from Ht−1 to Xti is measurable,

nonempty and compact valued, and sectionally continuous on Xt−1 × St−1.

The additional component of Nature is given by a measurable, nonempty

and closed valued correspondence At0 from Gr(At) to St, which is sectionally

continuous on Xt × St−1. Then Ht = Gr(At0)× St, and H∞ is the subset of

X∞ × S∞ such that (x, s) ∈ H∞ if (xt, st) ∈ Ht for any t ≥ 0.

3. The choices of Nature depend not only on the history ht−1, but also on the

action profile xt in the current stage. The state transition ft0(ht−1, xt) =

ft0(ht−1, xt) � ft0(ht−1, xt), where ft0 is a transition probability from Gr(At)

toM(St) such that ft0(At0(ht−1, xt)|ht−1, xt) = 1 for all (ht−1, xt) ∈ Gr(At),

and ft0 is a transition probability from Gr(At0) to M(St).

4. For each i ∈ I, the payoff function ui is a Borel measurable mapping from H∞

to R++ which is bounded by γ > 0, and sectionally continuous on X∞× S∞.

We allow the possibility for the players to have perfect information in some

stages. For t ≥ 1, let

Nt =

1, if ft0(ht−1, xt) ≡ δst for some st and

|{i ∈ I : Ati is not point valued}| = 1;

0, otherwise,

where |K| represents the number of points in the set K. Thus, if Nt = 1 for some

stage t, then the player who is active in the period t is the only active player and

has perfect information.

We will drop the ARM condition in those periods with only one active player,

and weaken the ARM condition in other periods.

Assumption 3 (ARM′). 1. For any t ≥ 1 with Nt = 1, St is a singleton set

{st} and λt = δst.

2. For each t ≥ 1 with Nt = 0, ft0 is sectionally continuous on Xt × St−1,

where the range space M(St) is endowed with topology of weak convergence

of measures on St. The probability measure ft0(·|ht−1, xt, st) is absolutely

continuous with respect to an atomless Borel probability measure λt on St

for all (ht−1, xt, st) ∈ Gr(At0), and ϕt0(ht−1, xt, st, st) is the corresponding

density.21

21In this subsection, a property is said to hold for λt-almost all ht ∈ Ht if it is satisfied for λt-almostall st ∈ St and all (xt, st) ∈ Ht(s

t).

12

3. The mapping ϕt0 is Borel measurable and sectionally continuous on Xt ×St, and integrably bounded in the sense that there is a λt-integrable function

φt : St → R+ such that ϕt0(ht−1, xt, st, st, ) ≤ φt(st) for any (ht−1, xt, st).

The following proposition shows that the existence result is still true in this

more general setting.

Proposition 2. If an infinite-horizon dynamic game satisfies the ARM′ condition

and is continuous at infinity, then it possesses a subgame-perfect equilibrium f . In

particular, for j ∈ I and t ≥ 1 such that Nt = 1 and player j is the only active

player in this period, ftj can be deterministic. Furthermore, the equilibrium payoff

correspondence Et is nonempty and compact valued, and essentially sectionally

upper hemicontinuous on Xt−1 × St−1.

Remark 2. The proposition above also implies a new existence result of subgame-

perfect equilibria for stochastic games. Consider a standard stochastic game

with uncountable states as in Mertens and Parthasarathy (2003). Mertens and

Parthasarathy (2003) proved the existence of a subgame-perfect equilibrium by

assuming the state transitions to be norm continuous with respect to the actions in

the previous stage. On the contrary, our Proposition 2 allows the state transitions

to have a weakly continuous component.

Dynamic games with perfect information is a special class of dynamic games

in which players move sequentially. As noted in Footnote 7, such games have

been extensively studied and found wide applications in economics. As an

immediate corollary, an equilibrium existence result for dynamic games with perfect

information is given below.

Corollary 2. If a dynamic game with perfect information satisfies the ARM′

condition and is continuous at infinity, then it possesses a pure-strategy subgame-

perfect equilibrium.

4.2 Continuous dynamic games with partially perfect

information

In this subsection, we will study an infinite-horizon dynamic game with a

continuous structure. As in the previous subsection, we allow the state transition

to depend on the action profile in the current stage as well as on the previous

history, and the players may have perfect information in some stages.

1. For each t ≥ 1, the choices of Nature depends not only on the history ht−1, but

also on the action profile xt in this stage. For any t ≥ 1, suppose that At0 is a

13

continuous, nonempty and closed valued correspondence from Gr(At) to St.

Then Ht = Gr(At0), and H∞ is the subset of X∞×S∞ such that (x, s) ∈ H∞if (xt, st) ∈ Ht for any t ≥ 0.

2. Nature’s action is given by a mapping ft0 from Gr(At) to M(St) such that

ft0(At0(ht−1, xt)|ht−1, xt) = 1 for all (ht−1, xt) ∈ Gr(At).

3. For each t ≥ 1, let

Nt =

1, if ft0(ht−1, xt) ≡ δst for some st and

|{i ∈ I : Ati is not point valued}| = 1;

0, otherwise.

Definition 3. A dynamic game is said to be continuous if for each t and i,

1. the action correspondence Ati is continuous on Ht−1;

2. the transition probability ft0 is a continuous mapping from Gr(At) to M(St),

where M(St) is endowed with topology of weak convergence of measures on

St;

3. the payoff function ui is continuous on H∞.

Note that the “continuity at infinity” condition is automatically satisfied in a

continuous dynamic game.

Next, we propose the condition of “atomless transitions” on the state space,

which means that the state transition is an atomless probability measure in any

stage. This condition is slightly weaker than the ARM condition.

Assumption 4 (Atomless Transitions). 1. For any t ≥ 1 with Nt = 1, St is a

singleton set {st}.

2. For each t ≥ 1 with Nt = 0, ft0(ht−1) is an atomless Borel probability measure

for each ht−1 ∈ Ht−1.

Since we work with continuous dynamic games, we can adopt a slightly stronger

notion of subgame-perfect equilibrium. That is, each player’s strategy is optimal

in every subgame given the strategies of all other players.

Definition 4 (SPE′). A subgame-perfect equilibrium is a strategy profile f such

that for all i ∈ I, t ≥ 0, and all ht ∈ Ht, player i cannot improve his payoff in the

subgame ht by a unilateral change in his strategy.

The result on the equilibrium existence is presented below.

14

Proposition 3. If a continuous dynamic game has atomless transitions, then it

possesses a subgame-perfect equilibrium f . In particular, for j ∈ I and t ≥ 1

such that Nt = 1 and player j is the only active player in this period, ftj can

be deterministic. In addition, Et is nonempty and compact valued, and upper

hemicontinuous on Ht−1 for any t ≥ 1.

Remark 3. Proposition 3 goes beyond the main result of Harris, Reny and Robson

(1995). They proved the existence of a subgame-perfect correlated equilibrium

in a continuous dynamic game with almost perfect information by introducing a

public randomization device, which does not influence the payoffs, transitions or

action correspondences. It is easy to see that their model automatically satisfies the

condition of atomless transitions. The state in our model is completely endogenous

in the sense that it affects all the model parameters such as payoffs, transitions,

and action correspondences.

Remark 4. Proposition 3 above provides a new existence result for continuous

stochastic games. As remarked in the previous subsection, the existence of subgame-

perfect equilibria has been proved for general stochastic games with a stronger

continuity assumption on the state transitions, namely the norm continuity. On

the contrary, we only need to require the state transitions to be weakly continuous.

Remark 5. The condition of atomless transitions is minimal. In particular, the

counterexample provided by Luttmer and Mariotti (2003), which is a continuous

dynamic game with perfect information and Nature, does not have any subgame-

perfect equilibrium. In their example, Nature is active in the third period, but the

state transitions could have atoms. Thus, our condition of atomless transitions is

violated.

The next corollary follows from Proposition 3, which presents the existence

result for continuous dynamic games with perfect information (and Nature).

Corollary 3. If a continuous dynamic game with perfect information has atomless

transitions, then it possesses a pure-strategy subgame-perfect equilibrium.

Remark 6. Harris (1985), Hellwig and Leininger (1987), Borgers (1989) and

Hellwig et al. (1990) proved the existence of subgame-perfect equilibria in contin-

uous dynamic games with perfect information. In particular, Nature is absent in

all those papers. Luttmer and Mariotti (2003) provided an example of a five-stage

continuous dynamic game with perfect information, in which Nature is present

and no subgame-perfect equilibrium exists. The only known general existence

result, to the best of our knowledge, for (finite or infinite horizon) continuous

dynamic games with perfect information and Nature is the existence of subgame-

perfect correlated equilibria via public randomization as in Harris, Reny and Robson

(1995). Corollary 3 covers all those existence results as special cases.

15

5 Appendix

In Subsection 5.1, we present Lemmas 1-15 as the mathematical preparations for

proving Theorem 1. Since correspondences will be used extensively in the proofs,

we collect, for the convenience of the reader, several known results on various

properties of correspondences as Lemmas 1-7. The proof for the finite-horizon case

of Theorem 1 will depend on Lemmas 1-7. The rest of the lemmas (Lemmas 8-15)

will only be used for proving the infinite-horizon case.

We present in Subsection 5.2 a new existence result for discontinuous games

with stochastic endogenous sharing rules. The proofs of Theorem 1 and Proposition

1 are given in Subsection 5.3. Finally, the proofs for the results in Section 4 are

provided in Subsections 5.4 and 5.5.

5.1 Technical preparations

Let (S,S) be a measurable space and X a topological space with its Borel σ-algebra

B(X). A correspondence Ψ from S to X is a function from S to the space of all

subsets of X. The upper inverse Ψu of a subset A ⊆ X is

Ψu(A) = {s ∈ S : Ψ(s) ⊆ A}.

The lower inverse Ψl of a subset A ⊆ X is

Ψl(A) = {s ∈ S : Ψ(s) ∩A 6= ∅}.

The correspondence Ψ is

1. weakly measurable, if Ψl(O) ∈ S for each open subset O ⊆ X;

2. measurable, if Ψl(K) ∈ S for each closed subset K ⊆ X.

The graph of Ψ is denoted by Gr(Ψ) = {(s, x) ∈ S × X : s ∈ S, x ∈ Ψ(s)}. The

correspondence Ψ is said to have a measurable graph if Gr(Ψ) ∈ S ⊗ B(X).

If S is a topological space, then Ψ is

1. upper hemicontinuous, if Ψu(O) is open for each open subset O ⊆ X;

2. lower hemicontinuous, if Ψl(O) is open for each open subset O ⊆ X.

The following lemma shows that the space of nonempty compact subsets of a

Polish space is still Polish under the Hausdorff metric topology.

Lemma 1. Suppose that X is a Polish space and K is the set of all nonempty

compact subsets of X endowed with the Hausdorff metric topology. Then K is a

Polish space.

16

Proof. By Theorem 3.88 (2) of Aliprantis and Border (2006), K is complete. In

addition, Corollary 3.90 and Theorem 3.91 of Aliprantis and Border (2006) imply

that K is separable. Thus, K is a Polish space.

The following two lemmas present some basic measurability and continuity

properties for correspondences.

Lemma 2. Let (S,S) be a measurable space, X a Polish space endowed with the

Borel σ-algebra B(X), and K the space of nonempty compact subsets of X endowed

with its Hausdorff metric topology. Suppose that Ψ: S → X is a nonempty and

closed valued correspondence.

1. If Ψ is weakly measurable, then it has a measurable graph.

2. If Ψ is compact valued, then the following statements are equivalent.

(a) The correspondence Ψ is weakly measurable.

(b) The correspondence Ψ is measurable.

(c) The function f : S → K, defined by f(s) = Ψ(s), is Borel measurable.

3. Suppose that S is a topological space. If Ψ is compact valued, then the

function f : S → K defined by f(s) = Ψ(s) is continuous if and only if the

correspondence Ψ is continuous.

4. Suppose that (S,S, λ) is a complete probability space. Then Ψ is S-measurable

if and only if it has a measurable graph.

5. For a correspondence Ψ: S → X between two Polish spaces, the following

statements are equivalent.

(a) The correspondence Ψ is upper hemicontinuous at a point s ∈ S and

Ψ(s) is compact.

(b) If a sequence (sn, xn) in the graph of Ψ satisfies sn → s, then the sequence

{xn} has a limit in Ψ(s).

6. For a correspondence Ψ: S → X between two Polish spaces, the following

statements are equivalent.

(a) The correspondence Ψ is lower hemicontinuous at a point s ∈ S.

(b) If sn → s, then for each x ∈ Ψ(s), there exist a subsequence {snk} of

{sn} and elements xk ∈ Ψ(snk) for each k such that xk → x.

7. Given correspondences F : X → Y and G : Y → Z, the composition F and G

is defined by

G(F (x)) = ∪y∈F (x)G(y).

17

The composition of upper hemicontinuous correspondences is upper hemicon-

tinuous. The composition of lower hemicontinuous correspondences is lower

hemicontinuous.

Proof. Properties (1), (2), (3), (5), (6) and (7) are Theorems 18.6, 18.10, 17.15,

17.20, 17.21 and 17.23 of Aliprantis and Border (2006), respectively. Property (4)

follows from Proposition 4 of Hildenbrand (1974, p.61).

Lemma 3. 1. A correspondence Ψ from a measurable space (S,S) into a topo-

logical space X is weakly measurable if and only if its closure correspondence

Ψ is weakly measurable, where for each s ∈ S, Ψ(s) = Ψ(s) and Ψ(s) is the

closure of the set Ψ(s) in X.

2. For a sequence {Ψm} of correspondences from a measurable space (S,S)

into a Polish space, the union correspondence Ψ(s) = ∪m≥1Ψm(s) is weakly

measurable if each Ψm is weakly measurable. If each Ψm is weakly measurable

and compact valued, then the intersection correspondence Φ(s) = ∩m≥1Ψm(s)

is weakly measurable.

3. A weakly measurable, nonempty and closed valued correspondence from a

measurable space into a Polish space admits a measurable selection.

4. A correspondence with closed graph between compact metric spaces is mea-

surable.

5. A nonempty and compact valued correspondence Ψ from a measurable space

(S,S) into a Polish space is weakly measurable if and only if there exists

a sequence {ψ1, ψ2. . . .} of measurable selections of Ψ such that Ψ(s) =

{ψ1(s), ψ2(s), . . .} for each s ∈ S.

6. The image of a compact set under a compact valued upper hemicontinuous

correspondence is compact.22 If the domain is compact, then the graph of a

compact valued upper hemicontinuous correspondence is compact.

7. The intersection of a correspondence with closed graph and an upper hemi-

continuous compact valued correspondence is upper hemicontinuous.

8. If the correspondence Ψ: S → Rl is compact valued and upper hemi-

continuous, then the convex hull of Ψ is also compact valued and upper

hemicontinuous.

Proof. Properties (1)-(7) are Lemmas 18.3 and 18.4, Theorems 18.13 and 18.20,

Corollary 18.15, Lemma 17.8 and Theorem 17.25 in Aliprantis and Border (2006),

respectively. Property 8 is Proposition 6 in Hildenbrand (1974, p.26).

22Given a correspondence F : X → Y and a subset A of X, the image of A under F is defined to bethe set ∪x∈AF (x).

18

Parts 1 and 2 of the following lemma are the standard Lusin’s Theorem

and Michael’s continuous selection theorem, while the other two parts are about

properties involving mixture of measurability and continuity of correspondences.

Lemma 4. 1. Lusin’s Theorem: Suppose that S is a Borel subset of a Polish

space, λ is a Borel probability measure on S and S is the completion of B(S)

under λ. Let X be a Polish space. If f is an S-measurable mapping from S to

X, then for any ε > 0, there exists a compact subset S1 ⊆ S with λ(S\S1) < ε

such that the restriction of f to S1 is continuous.

2. Michael’s continuous selection theorem: Let S be a metrizable space, and X a

complete metrizable closed subset of some locally convex space. Suppose that

F : S → X is a lower hemicontinuous, nonempty, convex and closed valued

correspondence. Then there exists a continuous mapping f : S → X such that

f(s) ∈ F (s) for all s ∈ S.

3. Let (S,S, λ) be a finite measure space, X a Polish space, and Y a locally

convex linear topological space. Let F : S → X be a closed-valued corre-

spondence such that Gr(F ) ∈ S ⊗ B(X), and f : Gr(F ) → Y a measurable

function which is sectionally continuous on X. Then there exists a measurable

function f ′ : S × X → Y such that (1) f ′ is sectionally continuous on

X, (2) for λ-almost all s ∈ S, f ′(s, x) = f(s, x) for all x ∈ F (s) and

f ′(s,X) ⊆ cof(s, F (s)).23

4. Let (S,S) be a measurable space, and X and Y Polish spaces. Let Ψ: S×X →M(Y ) be an S ⊗ B(X)-measurable, nonempty, convex and compact valued

correspondence which is sectionally continuous on X, where the compactness

and continuity are with respect to the weak∗ topology on M(Y ). Then there

exists an S⊗B(X)-measurable selection ψ of Ψ that is sectionally continuous

on X.

Proof. Lusin’s theorem is Theorem 7.1.13 in Bogachev (2007). Michael’s contin-

uous selection theorem can be found in Michael (1966) and the last paragraph of

Bogachev (2007, p.228). Properties (3) is Theorem 2.7 in Brown and Schreiber

(1989). Property (4) follows from Theorem 1 and the main lemma of Kucia

(1998).

The following lemma presents the convexity, compactness and continuity

properties on integration of correspondences.

23For any set A in a linear topological space, coA denotes the convex hull of A.

19

Lemma 5. Let (S,S, λ) be an atomless probability space, X a Polish space, and

F a correspondence from S to Rl. Denote∫SF (s)λ(ds) =

{∫Sf(s)λ(ds) : f is an integrable selection of F on S

}.

1. If F is measurable, nonempty and closed valued, and λ-integrably bounded by

some integrable function ψ : S → R+ in the sense that for λ-almost all s ∈ S,

‖y‖ ≤ ψ(s) for any y ∈ F (s), then∫S F (s)λ(ds) is nonempty, convex and

compact, and ∫SF (s)λ(ds) =

∫S

coF (s)λ(ds).

2. If G is a measurable, nonempty and closed valued correspondence from S ×X → Rl such that (1) G(s, ·) is upper (resp. lower) hemicontinuous on X

for all s ∈ S, and (2) G is λ-integrably bounded by some integrable function

ψ : S → R+ in the sense that for λ-almost all s ∈ S, ‖y‖ ≤ ψ(s) for any x ∈ Xand y ∈ G(s, x), then

∫S G(s, x)λ(ds) is upper (resp. lower) hemicontinuous

on X.

Proof. See Theorems 2, 3 and 4, Propositions 7 and 8, and Problem 6 in

Section D.II.4 of Hildenbrand (1974).

The following result proves a measurable version of Lyapunov’s theorem, which

is taken from Mertens (2003). Let (S,S) and (X,X ) be measurable spaces. A

transition probability from S to X is a mapping f from S to the space M(X) of

probability measures on (X,X ) such that f(B|·) : s→ f(B|s) is S-measurable for

each B ∈ X .

Lemma 6. Let f(·|s) be a transition probability from a measurable space (S,S)

to another measurable space (X,X ) (X is separable).24 Let Q be a measurable,

nonempty and compact valued correspondence from S × X to Rl, which is f -

integrable in the sense that for any measurable selection q of Q, q(·, s) is f(·|s)-absolutely integrable for any s ∈ S. Let

∫Qdf be the correspondence from S to

subsets of Rl defined by

M(s) =

(∫Qdf

)(s) =

{∫Xq(s, x)f(dx|s) : q is a measurable selection of Q

}.

Denote the graph of M by J . Let J be the restriction of the product σ-algebra

S ⊗ B(Rl) to J .

Then

24A σ-algebra is said to be separable if it is generated by a countable collection of sets.

20

1. M is a measurable, nonempty and compact valued correspondence;

2. there exists a measurable, Rl-valued function g on (X × J,X ⊗ J ) such that

g(x, e, s) ∈ Q(x, s) and e =∫X g(x, e, s)f(dx|s).

The following result is Lemma 6 of Reny and Robson (2002).

Lemma 7. Suppose that H and X are compact metric spaces. Let P : H×X → Rn

be a nonempty valued and upper hemicontinuous correspondence, and the mappings

f : H → M(X) and µ : H → 4(X) be measurable. In addition, suppose that

µ(·|h) = p(h, ·) ◦ f(·|h) such that p(h, ·) is a measurable selection of P (h, ·). Then

there exists a jointly Borel measurable selection g of P such that µ(·|h) = g(h, ·) ◦f(·|h); that is, g(h, x) = p(h, x) for f(·|h)-almost all x.

Suppose that (S1,S1) is a measurable space, S2 is a Polish space endowed with

the Borel σ-algebra, and S = S1×S2 which is endowed with the product σ-algebra

S. Let D be an S-measurable subset of S such that D(s1) is compact for any

s1 ∈ S1. The σ-algebra D is the restriction of S on D. Let X be a Polish space,

and A a D-measurable, nonempty and closed valued correspondence from D to X

which is sectionally continuous on S2. The following lemma considers the property

of upper hemicontinuity for the correspondence M as defined in Lemma 6.

Lemma 8. Let f(·|s) be a transition probability from (D,D) to M(X) such that

f(A(s)|s) = 1 for any s ∈ D, which is sectionally continuous on S2. Let G be a

bounded, measurable, nonempty, convex and compact valued correspondence from

Gr(A) to Rl, which is sectionally upper hemicontinuous on S2 ×X. Let∫Gdf be

the correspondence from D to subsets of Rl defined by

M(s) =

(∫G df

)(s) =

{∫Xg(s, x)f(dx|s) : g is a measurable selection of G

}.

Then M is S-measurable, nonempty and compact valued, and sectionally upper

hemicontinuous on S2.

Proof. Define a correspondence G : S ×X → Rl as

G =

G(s, x), if (s, x) ∈ Gr(A);

{0}, otherwise.

Then M(s) =(∫

G df)

(s) =(∫G df

)(s). The measurability, nonemptiness and

compactness follows from Lemma 6. Given s1 ∈ S1 such that (1) D(s1) 6= ∅,(2) f(s1, ·) and G(s1, ·, ·) is upper hemicontinuous. The upper hemicontinuity of

M(s1, ·) follows from Lemma 2 in Simon and Zame (1990) and Lemma 4 in Reny

and Robson (2002).

21

From now onwards, whenever we work with mappings taking values in the

spaceM(X) of Borel probability measures on some separable metric space X, the

relevant continuity or convergence is assumed to be in terms of the topology of

weak convergence of measures on M(X) unless otherwise noted.

In the following lemma, we state some properties for transition correspondences.

Lemma 9. Suppose that Y and Z are Polish spaces. Let G be a measurable,

nonempty, convex and compact valued correspondence from Y to M(Z). Define a

correspondence G′ from M(Y ) to M(Z) as

G′(ν) =

{∫Yg(y)ν(dy) : g is a Borel measurable selection of G

}.25

1. The correspondence G′ is measurable, nonempty, convex and compact valued.

2. The correspondence G is upper hemicontinuous if and only if G′ is upper

hemicontinuous. In addition, if G is continuous, then G′ is continuous.

Proof. (1) is Lemma 19.29 of Aliprantis and Border (2006). By Theorem 19.30

therein, G is upper hemicontinuous if and only if G′ is upper hemicontinuous. We

need to show that G′ is lower hemicontinuous if G is lower hemicontinuous.

Let Z be endowed with a totally bounded metric, and U(Z) the space of

bounded, real-valued and uniformly continuous functions on Z endowed with the

supremum norm, which is obviously separable. Pick a countable set {fm}m≥1 ⊆U(Z) such that {fm} is dense in the unit ball of U(Z). It follows from Theorem 15.2

of Aliprantis and Border (2006) that the weak∗ topology ofM(Z) is metrizable by

the metric dz, where

dz(µ1, µ2) =∞∑m=1

1

2m

∣∣∣∣∫Zfm(z)µ1(dz)−

∫Zfm(z)µ2(dz)

∣∣∣∣for each pair of µ1, µ2 ∈M(Z).

Suppose that {νj}j≥0 is a sequence in M(Y ) such that νj → ν0 as j → ∞.

Pick an arbitrary point µ0 ∈ G′(ν0). By the definition of G′, there exists a Borel

measurable selection g of G such that µ0 =∫Y g(y)ν0(dy).

For each k ≥ 1, by Lemma 4 (Lusin’s theorem), there exists a compact subset

Dk ⊆ Y such that g is continuous on Dk and ν0(Y \ Dk) < 13k . Define a

25The integral∫Yg(y)ν(dy) defines a Borel probability measure τ on Z such that for any Borel set C

in Z, τ(C) =∫Yg(y)(C)ν(dy). The measure τ is also equal to the Gelfand integral of g with respect to

the measure ν on Y , whenM(Z) is viewed as a set in the dual space of the space of bounded continuousfunctions on Z; see Definition 11.49 of Aliprantis and Border (2006).

22

correspondence Gk : Y →M(X) as follows:

Gk(y) =

{g(y)}, y ∈ Dk;

G(y), y ∈ Y \Dk.

Then Gk is nonempty, convex and compact valued, and lower hemicontinuous.

By Theorem 3.22 in Aliprantis and Border (2006), Y is paracompact. Then by

Lemma 4 (Michael’s selection theorem), it has a continuous selection gk.

For each k, since νj → ν0 and gk is continuous,∫Y gk(y)νj(dy)→

∫Y gk(y)ν0(dy)

in the sense that for any m ≥ 1,∫Y

∫Zfm(z)gk(dz|y)νj(dy)→

∫Y

∫Zfm(z)gk(dz|y)ν0(dy).

Thus, there exists a point νjk such that {jk} is an increasing sequence and

dz

(∫Ygk(y)νjk(dy),

∫Ygk(y)ν0(dy)

)<

1

3k.

In addition, since gk coincides with g on Dk and ν0(Y \Dk) <13k ,

dz

(∫Ygk(y)ν0(dy),

∫Yg(y)ν0(dy)

)<

2

3k.

Thus,

dz

(∫Ygk(y)νjk(dy),

∫Yg(y)ν0(dy)

)<

1

k.

Let µjk =∫Y gk(y)νjk(dy) for each k. Then µjk ∈ G′(νjk) and µjk → µ0 as k →∞.

By Lemma 2, G′ is lower hemicontinuous.

The next lemma presents some properties for the composition of two transition

correspondences in terms of the product of transition probabilities.

Lemma 10. Let X, Y and Z be Polish spaces, and G a measurable, nonempty and

compact valued correspondence from X to M(Y ). Suppose that F is a measurable,

nonempty, convex and compact valued correspondence from X×Y toM(Z). Define

a correspondence Π from X to M(Y × Z) as follows:

Π(x) = {g(x) � f(x) : g is a Borel measurable selection of G,

f is a Borel measurable selection of F}.

1. If F is sectionally continuous on Y , then Π is a measurable, nonempty and

compact valued correspondence.

23

2. If there exists a function g from X toM(Y ) such that G(x) = {g(x)} for any

x ∈ X, then Π is a measurable, nonempty and compact valued correspondence.

3. If both G and F are continuous correspondences, then Π is a nonempty and

compact valued, and continuous correspondence.26

4. If G(x) ≡ {λ} for some fixed Borel probability measure λ ∈ M(Y ) and F is

sectionally continuous on X, then Π is a continuous, nonempty and compact

valued correspondence.

Proof. (1) Define three correspondences F : X×Y →M(Y ×Z), F : M(X×Y )→M(Y × Z) and F : X ×M(Y )→M(Y × Z) as follows:

F (x, y) = {δy ⊗ µ : µ ∈ F (x, y)},

F (τ) =

{∫X×Y

f(x, y)τ(d(x, y)) : f is a Borel measurable selection of F

},

F (x, µ) = F (δx ⊗ µ).

Since F is measurable, nonempty, convex and compact valued, F is measurable,

nonempty, convex and compact valued. By Lemma 9, the correspondence F is

measurable, nonempty, convex and compact valued, and F (x, ·) is continuous on

M(Y ) for any x ∈ X.

Since G is measurable and compact valued, there exists a sequence of Borel

measurable selections {gk}k≥1 of G such that G(x) = {g1(x), g2(x), . . .} for any

x ∈ X by Lemma 3 (5). For each k ≥ 1, define a correspondence Πk from X to

M(Y ×Z) by letting Πk(x) = F (x, gk(x)) = F (x⊗gk(x)). Then Πk is measurable,

nonempty, convex and compact valued.

Fix any x ∈ X. It is clear that Π(x) = F (x,G(x)) is a nonempty valued. Since

G(x) is compact, and F (x, ·) is compact valued and continuous, Π(x) is compact

by Lemma 3. Thus, the closure⋃k≥1 Πk(x) of

⋃k≥1 Πk(x) is a subset of Π(x).

Fix any x ∈ X and τ ∈ Π(x). There exists a point ν ∈ G(x) such that

τ ∈ F (x, ν). Since {gk(x)}k≥1 is dense in G(x), it has a subsequence {gkm(x)}such that gkm(x)→ ν. As F (x, ·) is continuous, F (x, gkm(x))→ F (x, ν). That is,

τ ∈⋃k≥1

F (x, gk(x)) =⋃k≥1

Πk(x).

Therefore,⋃k≥1 Πk(x) = Π(x) for any x ∈ X. Lemma 3 (1) and (2) imply that Π

is measurable.

26In Lemma 29 of Harris, Reny and Robson (1995), they showed that Π is upper hemicontinuous ifboth G and F are upper hemicontinuous.

24

(2) As in (1), the correspondence F is measurable, nonempty, convex and

compact valued. If G = {g} for some measurable function g, then Π(x) =

F (δx ⊗ g(x)), which is measurable, nonempty and compact valued.

(3) We continue to work with the two correspondences F : X×Y →M(Y ×Z)

and F : M(X×Y )→M(Y ×Z) as in Part (1). By the condition on F , it is obvious

that the correspondence F is continuous, nonempty, convex and compact valued.

Lemma 9 implies the properties for the correspondence F . Define a correspondence

G : X → M(X × Y ) as G(x) = δx ⊗ G(x). Since G and F are both nonempty

valued, Π(x) = F (G(x)) is nonempty. As G is compact valued and F is continuous,

Π is compact valued by Lemma 3. As G and F are both continuous, Π is continuous

by Lemma 2 (7).

(4) Let Y ′ = Y and define a correspondence F : X×Y →M(Y ′×Z) as follows:

F (x, y) = {δy ⊗ µ : µ ∈ F (x, y)}.

Then F is also measurable, nonempty, convex and compact valued, and sectionally

upper hemicontinuous on X.

Let d be a totally bounded metric on Y ′ × Z, and U(Y ′ × Z) the space of

bounded, real-valued and uniformly continuous functions on Y ′×Z endowed with

the supremum norm, which is obviously separable. It follows from Theorem 15.2 of

Aliprantis and Border (2006) that the space of Borel probability measures on Y ′×Zwith the topology of weak convergence of measures can be viewed as a subspace

of the dual space of U(Y ′ × Z) with the weak* topology. By Corollary 18.37 of

Aliprantis and Border (2006), Π(x) =∫Y F (x, y)λ(dy) is nonempty, convex and

compact for any x ∈ X.27

Now we shall show the upper hemicontinuity. If xn → x0 and µn ∈ Π(xn), we

need to prove that there exists some µ0 ∈ Π(x0) such that a subsequence of {µn}weakly converges to µ0. Suppose that for n ≥ 1, fn is a Borel measurable selection

of F (xn, ·) such that µn = λ � fn.

Fix any y ∈ Y , let J(y) = co{fn(y)⊗δxn}n≥1, which is the closure of the convex

hull of {fn(y)⊗δxn}n≥1. It is obvious that J(y) is nonempty and convex. It is also

clear that J(y) is the closure of the countable set{n∑i=1

αifi(y)⊗ δxi : n ≥ 1, αi ∈ Q+, i = 1, . . . , n,

n∑i=1

αi = 1

},

27Note that the integral∫YF (x, y)λ(dy) can be viewed as the Gelfand integral of the correspondence

in the dual space of U(Y ′ × Z); see Definition 18.36 of Aliprantis and Border (2006).

25

where Q+ is the set of non-negative rational numbers. Let F ′(x) = {µ ⊗ δx :

µ ∈ F (x, y)} for any x ∈ X. Then, F ′ is continuous on X. Since {xn : n ≥0} is a compact set, Lemma 3 (6) implies that

⋃n≥0 F

′(xn) is compact. Hence,

{fn(y)⊗ δxn}n≥1 is relatively compact. By Theorem 5.22 of Aliprantis and Border

(2006), {fn(y)⊗δxn}n≥1 is tight. That is, for any positive real number ε, there is a

compact set Kε in Z ×X such that for any n ≥ 1, fn(y)⊗ δxn (Kε) > 1− ε. Thus,∑ni=1 αifi(y) ⊗ δxi (Kε) > 1 − ε for any n ≥ 1, and for any αi ∈ Q+, i = 1, . . . , n

with∑n

i=1 αi = 1. Hence, J(y) is compact by Theorem 5.22 of Aliprantis and

Border (2006) again.

For any n ≥ 1, and for any αi ∈ Q+, i = 1, . . . , n with∑n

i=1 αi = 1, it is clear

that∑n

i=1 αifi(y) ⊗ δxi is measurable in y ∈ Y . Lemma 3 (5) implies that J is

also a measurable correspondence from Y to M(Z ×X). By the argument in the

second paragraph of the proof of Part (4), the set

Λ = {λ � ζ : ζ is a Borel measurable selection of J}

is compact.

Since λ�(fn⊗δxn) ∈ Λ for each n, there exists some Borel measurable selection

ζ of J such that a subsequence of {λ � (fn ⊗ δxn)}, say itself, weakly converges to

λ � ζ ∈ Λ. Let ζX(y) be the marginal probability of ζ(y) on X for each y. Since xn

converges to x0, ζX(y) = δx0 for λ-almost all y ∈ Y . As a result, there exists a Borel

measurable function f0 such that ζ = f0 ⊗ δx0 , where f0(y) ∈ coLsn{fn(y)} for λ-

almost all y ∈ Y . Since F is convex and compact valued, and upper hemicontinuous

on X, f0 is a measurable selection of F (x0, ·). Let µ0 = λ � f0. Then µn weakly

converges to µ0, which implies that Π is upper hemicontinuous.

Next we shall show the lower hemicontinuity of Π. Suppose that xn → x0 and

µ0 ∈ Π(x0). Then there exists a Borel measurable selection f0 of F (x0, ·) such that

µ0 = λ�f0. Since F is sectionally lower hemicontinuous on X and compact valued,

for each n ≥ 1, there exists a measurable selection fn for F (xn, ·) such that fn(y)

weakly converges to f0(y) for each y ∈ Y .28 Denote µn = λ � fn. For any bounded

continuous function ψ on Y ×Z,∫Z ψ(y, z)fn(dz|y) converges to

∫Z ψ(y, z)f0(dz|y)

for any y ∈ Y . Thus, we have∫Y×Z

ψ(y, z)µn(d(y, z)) =

∫Y

∫Zψ(y, z)fn(dz|y)λ(dy)

→∫Y

∫Zψ(y, z)f0(dz|y)λ(dy)

by the Dominated Convergence Theorem. Therefore, Π is lower hemicontinuous.

28See Proposition 4.2 in Sun (1997). Notice that the atomless Loeb probability measurable spaceassumption is not needed for the result of lower hemicontinuity as in Theorem 10 therein.

26

The proof is thus complete.

The following result presents a variant of Lemma 6 in terms of transition

correspondences.

Lemma 11. Let X and Y be Polish spaces, and Z a compact subset of Rl+.

Let G be a measurable, nonempty and compact valued correspondence from X to

M(Y ). Suppose that F is a measurable, nonempty, convex and compact valued

correspondence from X × Y to Z. Define a correspondence Π from X to Z as

follows:

Π(x) = {∫Yf(x, y)g(dy|x) : g is a Borel measurable selection of G,

f is a Borel measurable selection of F}.

If F is sectionally continuous on Y , then

1. the correspondence F : X × M(Y ) → Z as F (x, ν) =∫Y F (x, y)ν(dy) is

sectionally continuous on M(Y ); and

2. Π is a measurable, nonempty and compact valued correspondence.

3. If F and G are both continuous, then Π is continuous.

Proof. (1) For any fixed x ∈ X, the upper hemicontinuity of F (x, ·) follows from

Lemma 8.

Next, we shall show the lower hemicontinuity. Fix any x ∈ X. Suppose that

{νj}j≥0 is a sequence in M(Y ) such that νj → ν0 as j → ∞. Pick an arbitrary

point z0 ∈ F (x, ν0). Then there exists a Borel measurable selection f of F (x, ·)such that z0 =

∫Y f(y)ν0(dy).

By Lemma 4 (Lusin’s theorem), for each k ≥ 1, there exists a compact subset

Dk ⊆ Y such that f is continuous on Dk and ν0(Y \Dk) <1

3kM , where M > 0 is

the bound of Z. Define a correspondence Fk : Y → Z as follows:

Fk(y) =

{f(y)}, y ∈ Dk;

F (x, y), y ∈ Y \Dk.

Then Fk is nonempty, convex and compact valued, and lower hemicontinuous.

By Theorem 3.22 in Aliprantis and Border (2006), Y is paracompact. Then by

Lemma 4 (Michael’s selection theorem), Fk has a continuous selection fk.

For each k, since νj → ν0 and fk is bounded and continuous,∫Y fk(y)νj(dy)→∫

Y fk(y)ν0(dy) as j → ∞. Thus, there exists a subsequence {νjk} such that {jk}

27

is an increasing sequence, and for each k ≥ 1,∥∥∥∥∫Yfk(y)νjk(dy)−

∫Yfk(y)ν0(dy)

∥∥∥∥ < 1

3k,

where ‖ · ‖ is the Euclidean norm on Rl.Since fk coincides with f on Dk, ν0(Y \Dk) <

13kM , and Z is bounded by M ,∥∥∥∥∫

Yfk(y)ν0(dy)−

∫Yf(y)ν0(dy)

∥∥∥∥ < 2

3k.

Thus, ∥∥∥∥∫Yfk(y)νjk(dy)−

∫Yf(y)ν0(dy)

∥∥∥∥ < 1

k.

Let zjk =∫Y fk(y)νjk(dy) for each k. Then zjk ∈ F (x, νjk) and zjk → z0 as k →∞.

By Lemma 2, F (x, ·) is lower hemicontinuous.

(2) Since G is measurable and compact valued, there exists a sequence of Borel

measurable selections {gk}k≥1 of G such that G(x) = {g1(x), g2(x), . . .} for any

x ∈ X by Lemma 3 (5). For each k ≥ 1, define a correspondence Πk from X to Z

by letting Πk(x) = F (x, gk(x)) =∫Y F (x, y)gk(dy|x). Since F is convex valued, so

is Πk. By Lemma 6, Πk is also measurable, nonempty and compact valued.

Fix any x ∈ X. It is clear that Π(x) = F (x,G(x)) is nonempty valued. Since

G(x) is compact, and F (x, ·) is compact valued and continuous, Π(x) is compact

by Lemma 3. Thus,⋃k≥1 Πk(x) ⊆ Π(x).

Fix any x ∈ X and z ∈ Π(x). There exists a point ν ∈ G(x) such that

z ∈ F (x, ν). Since {gk(x)}k≥1 is dense in G(x), it has a subsequence {gkm(x)}such that gkm(x)→ ν. As F (x, ·) is continuous, F (x, gkm(x))→ F (x, ν). That is,

z ∈⋃k≥1

F (x, gk(x)) =⋃k≥1

Πk(x).

Therefore,⋃k≥1 Πk(x) = Π(x) for any x ∈ X. Lemma 3 (1) and (2) imply that Π

is measurable.

(3) Define a correspondence F : M(X × Y )→ Z as follows:

F (τ) =

{∫X×Y

f(x, y)τ(d(x, y)) : f is a Borel measurable selection of F

}.

By (1), F is continuous. Define a correspondence G : X →M(X × Y ) as G(x) =

{δx⊗ ν : ν ∈ G(x)}. Since G and F are both nonempty valued, Π(x) = F (G(x)) is

28

nonempty. As G is compact valued and F is continuous, Π is compact valued by

Lemma 3. As G and F are both continuous, Π is continuous by Lemma 2 (7).

The following lemma shows that a measurable and sectionally continuous

correspondence on a product space is approximately continuous on the product

space.

Lemma 12. Let S, X and Y be Polish spaces endowed with the Borel σ-algebras,

and λ a Borel probability measure on S. Denote S as the completion of the Borel

σ-algebra B(S) of S under the probability measure λ. Suppose that D is a B(S)⊗B(Y )-measurable subset of S × Y , where D(s) is nonempty and compact for all

s ∈ S. Let A be a nonempty and compact valued correspondence from D to X,

which is sectionally continuous on Y and has a B(S × Y ×X)-measurable graph.

Then

(i) A(s) = Gr(A(s, ·)) is an S-measurable mapping from S to the set of nonempty

and compact subsets KY×X of Y ×X;

(ii) there exist countably many disjoint compact subsets {Sm}m≥1 of S such that

(1) λ(∪m≥1Sm) = 1, and (2) for each m ≥ 1, Dm = D∩(Sm×Y ) is compact,

and A is nonempty and compact valued, and continuous on each Dm.

Proof. (i)A(s, ·) is continuous andD(s) is compact, Gr(A(s, ·)) ⊆ Y×X is compact

by Lemma 3. Thus, A is nonempty and compact valued. Since A has a measurable

graph, A is an S-measurable mapping from S to the set of nonempty and compact

subsets KY×X of Y ×X by Lemma 2 (4).

(ii) Define a correspondence D from S to Y such that D(s) = {y ∈ Y : (s, y) ∈D}. Then D is nonempty and compact valued. As in (i), D is S-measurable.

By Lemma 4 (Lusin’s Theorem), there exists a compact subset S1 ⊆ S such that

λ(S \ S1) < 12 , D and A are continuous functions on S1. By Lemma 2 (3), D and

A are continuous correspondences on S1. Let D1 = {(s, y) ∈ D : s ∈ S1, y ∈ D(s)}.Since S1 is compact and D is continuous, D1 is compact (see Lemma 3 (6)).

Following the same procedure, for any m ≥ 1, there exists a compact subset

Sm ⊆ S such that (1) Sm∩(∪1≤k≤m−1Sk) = ∅ and Dm = D∩(Sm×Y ) is compact,

(2) λ(Sm) > 0 and λ (S \ (∪1≤k≤mSm)) < 12m , and (3) A is nonempty and compact

valued, and continuous on Dm. This completes the proof.

The lemma below states an equivalence property for the weak convergence of

Borel probability measures obtained from the product of transition probabilities.

Lemma 13. Let S and X be Polish spaces, and λ a Borel probability measure on

S. Suppose that {Sk}k≥1 is a sequence of disjoint compact subsets of S such that

λ(∪k≥1Sk) = 1. For each k, define a probability measure on Sk as λk(D) = λ(D)λ(Sk)

29

for any measurable subset D ⊆ Sk. Let {νm}m≥0 be a sequence of transition

probabilities from S to M(X), and τm = λ � νm for any m ≥ 0. Then τm weakly

converges to τ0 if and only if λk � νm weakly converges to λk � ν0 for each k ≥ 1.

Proof. First, we assume that τm weakly converges to τ0. For any closed subset

E ⊆ Sk×X, we have lim supm→∞ τm(E) ≤ τ0(E). That is, lim supm→∞ λ�νm(E) ≤λ � ν0(E). For any k, 1

λ(Sk) lim supm→∞ λ � νm(E) ≤ 1λ(Sk)λ � ν0(E), which implies

that lim supm→∞ λk � νm(E) ≤ λk � ν0(E). Thus, λk � νm weakly converges to

λk � ν0 for each k ≥ 1.

Second, we consider the case that λk � νm weakly converges to λk � ν0 for each

k ≥ 1. For any closed subset E ⊆ S×X, let Ek = E∩(Sk×X) for each k ≥ 1. Then

{Ek} are disjoint closed subsets and lim supm→∞ λk � νm(Ek) ≤ λk � ν0(Ek). Since

λk � νm(E′) = 1λ(Sk)λ � νm(E′) for any k, m and measurable subset E′ ⊆ Sk ×X,

we have that lim supm→∞ λ � νm(Ek) ≤ λ � ν0(Ek). Thus,∑k≥1

lim supm→∞

λ � νm(Ek) ≤∑k≥1

λ � ν0(Ek) = λ � ν0(E).

Since the limit superior is subadditive, we have∑k≥1

lim supm→∞

λ � νm(Ek) ≥ lim supm→∞

∑k≥1

λ � νm(Ek) = lim supm→∞

λ � νm(E).

Therefore, lim supm→∞ λ � νm(E) ≤ λ � ν0(E), which implies that τm weakly

converges to τ0.

The following is a key lemma that allows one to drop the continuity condition

on the state variables through a reference measure in Theorem 1.

Lemma 14. Suppose that X,Y and S are Polish spaces and Z is a compact metric

space. Let λ be a Borel probability measure on S, and A a nonempty and compact

valued correspondence from Z ×S to X which is sectionally upper hemicontinuous

on Z and has a B(Z×S×X)-measurable graph. Let G be a nonempty and compact

valued, continuous correspondence from Z to M(X ×S). We assume that for any

z ∈ Z and τ ∈ G(z), the marginal of τ on S is λ and τ(Gr(A(z, ·))) = 1. Let F be a

measurable, nonempty, convex and compact valued correspondence from Gr(A)→M(Y ) such that F is sectionally continuous on Z ×X. Define a correspondence

Π from Z to M(X × S × Y ) by letting

Π(z) = {g(z) � f(z, ·) : g is a Borel measurable selection of G,

f is a Borel measurable selection of F}.

Then the correspondence Π is nonempty and compact valued, and continuous.

30

Proof. Let S be the completion of B(S) under the probability measure λ. By

Lemma 12, A(s) = Gr(A(s, ·)) can be viewed as an S-measurable mapping from

S to the set of nonempty and compact subsets KZ×X of Z × X. For any s ∈ S,

the correspondence Fs = F (·, s) is continuous on A(s). By Lemma 4, there exists

a measurable, nonempty and compact valued correspondence F from Z ×X × StoM(Y ) and a Borel measurable subset S′ of S with λ(S′) = 1 such that for each

s ∈ S′, Fs is continuous on Z ×X, and the restriction of Fs to A(s) is Fs.

By Lemma 4 (Lusin’s theorem), there exists a compact subset S1 ⊆ S′ such

that A is continuous on S1 and λ(S1) > 12 . Let K1 = A(S1). Then K1 ⊆ Z ×X is

compact.

Let C(K1,KM(Y )) be the space of continuous functions from K1 to KM(Y ),

where KM(Y ) is the set of nonempty and compact subsets of M(Y ). Suppose

that the restriction of S on S1 is S1. Let F1 be the restriction of F to K1 × S1.

Then F1 can be viewed as an S1-measurable function from S1 to C(K1,KM(Y ))

(see Theorem 4.55 in Aliprantis and Border (2006)). Again by Lemma 4 (Lusin’s

theorem), there exists a compact subset of S1, say itself, such that λ(S1) > 12

and F1 is continuous on S1. As a result, F1 is a continuous correspondence on

Gr(A) ∩ (S1 × Z ×X), so is F . Let λ1 be a probability measure on S1 such that

λ1(D) = λ(D)λ(S1) for any measurable subset D ⊆ S1.

For any z ∈ Z and τ ∈ G(z), the definition of G implies that there exists a

transition probability ν from S to X such that λ � ν = τ . Define a correspondence

G1 from Z toM(X×S) as follows: for any z ∈ Z, G1(z) is the set of all τ1 = λ1�νsuch that τ = λ � ν ∈ G(z). It can be easily checked that G1 is also a nonempty

and compact valued, and continuous correspondence. Let

Π1(z) = {τ1 � f(z, ·) : τ1 = λ1 � ν ∈ G1(z),

f is a Borel measurable selection of F}.

By Lemma 10, Π1 is nonempty and compact valued, and continuous. Furthermore,

it is easy to see that for any z, Π1(z) coincides with the set

{(λ1 � ν) � f(z, ·) : λ � ν ∈ G(z), f is a Borel measurable selection of F}.

Repeat this procedure, one can find a sequence of compact subsets {St} such

that (1) for any t ≥ 1, St ⊆ S′, St ∩ (S1 ∪ . . . St−1) = ∅ and λ(S1 ∪ . . .∪ St) ≥ tt+1 ,

(2) F is continuous on Gr(A)∩(St×Z×X), λt is a probability measure on St such

that λt(D) = λ(D)λ(St)

for any measurable subset D ⊆ St, and (3) the correspondence

Πt(z) = {(λt � ν) � f(z, ·) : λ � ν ∈ G(z),

31

f is a Borel measurable selection of F}.

is nonempty and compact valued, and continuous.

Pick a sequence {zk}, {νk} and {fk} such that (λ�νk)�fk(zk, ·) ∈ Π(zk), zk → z0

and (λ � νk) � fk(zk, ·) weakly converges to some κ. It is easy to see that (λt � νk) �fk(zk, ·) ∈ Πt(zk) for each t. As Π1 is compact valued and continuous, it has a

subsequence, say itself, such that zk converges to some z0 ∈ Z and (λ1�νk)�fk(zk, ·)weakly converges to some (λ1 �µ1) � f1(z0, ·) ∈ Π1(z0). Repeat this procedure, one

can get a sequence of {µm} and fm. Let µ(s) = µm(s) and f(z0, s, x) = fm(z0, s, x)

for any x ∈ A(z0, s) when s ∈ Sm. By Lemma 13, (λ � µ) � f(z0, ·) = κ, which

implies that Π is upper hemicontinuous.

Similarly, the compactness and lower hemicontinuity of Π follow from the

compactness and lower hemicontinuity of Πt for each t.

The next lemma presents the convergence property for the integrals of a

sequence of functions and probability measures.

Lemma 15. Let S and X be Polish spaces, and A a measurable, nonempty and

compact valued correspondence from S to X. Suppose that λ is a Borel probability

measure on S and {νn}1≤n≤∞ is a sequence of transition probabilities from S to

M(X) such that νn(A(s)|s) = 1 for each s and n. For each n ≥ 1, let τn = λ � νn.

Assume that the sequence {τn} of Borel probability measures on S ×X converges

weakly to a Borel probability measure τ∞ on S ×X. Let {gn}1≤n≤∞ be a sequence

of functions satisfying the following three properties.

1. For each n between 1 and ∞, gn : S ×X → R+ is measurable and sectionally

continuous on X.

2. For any s ∈ S and any sequence xn → x∞ in X, gn(s, xn) → g∞(s, x∞) as

n→∞.

3. The sequence {gn}1≤n≤∞ is integrably bounded in the sense that there exists

a λ-integrable function ψ : S → R+ such that for any n, s and x, gn(s, x) ≤ψ(s).

Then we have ∫S×X

gn(s, x)τn(d(s, x))→∫S×X

g∞(s, x)τ∞(d(s, x)).

Proof. By Theorem 2.1.3 in Castaing, De Fitte and Valadier (2004), for any

integrably bounded function g : S × X → R+ which is sectionally continuous on

X, we have ∫S×X

g(s, x)τn(d(s, x))→∫S×X

g(s, x)τ∞(d(s, x)). (2)

32

Let {yn}1≤n≤∞ be a sequence such that yn = 1n and y∞ = 0. Then yn → y∞.

Define a mapping g from S×X×{y1, . . . , y∞} such that g(s, x, yn) = gn(s, x). Then

g is measurable on S and continuous on X×{y1, . . . , y∞}. Define a correspondence

G from S to X × {y1, . . . , y∞} × R+ such that

G(s) = {(x, yn, c) : c ∈ g(s, x, yn), x ∈ A(s), 1 ≤ n ≤ ∞} .

For any s, A(s) × {y1, . . . , y∞} is compact and g(s, ·, ·) is continuous. By

Lemma 3 (6), G(s) is compact. By Lemma 2 (2), G can be viewed as a measurable

mapping from S to the space of nonempty compact subsets of X ×{y1, . . . , y∞}×R+. Similarly, A can be viewed as a measurable mapping from S to the space of

nonempty compact subsets of X.

Fix an arbitrary ε > 0. By Lemma 4 (Lusin’s theorem), there exists a compact

subset S1 ⊆ S such that A and G are continuous on S1 and λ(S \S1) < ε. Without

loss of generality, we can assume that λ(S \ S1) is sufficiently small such that∫S\S1

ψ(s)λ(ds) < ε6 . Thus, for any n,∫

(S\S1)×Xψ(s)τn(d(s, x)) =

∫(S\S1)

ψ(s)νn(X)λ(ds) <ε

6.

By Lemma 3 (6), the set E = {(s, x) : s ∈ S1, x ∈ A(s)} is compact. Since G

is continuous on S1, g is continuous on E × {y1, . . . , y∞}. Since E × {y1, . . . , y∞}is compact, g is uniformly continuous on E × {y1, . . . , y∞}. Thus, there exists a

positive integer N1 > 0 such that for any n ≥ N1, |gn(s, x)− g∞(s, x)| < ε3 for any

(s, x) ∈ E.

By Equation (2), there exists a positive integer N2 such that for any n ≥ N2,∣∣∣∣∫S×X

g∞(s, x)τn(d(s, x))−∫S×X

g∞(s, x)τ∞(d(s, x))

∣∣∣∣ < ε

3.

Let N0 = max{N1, N2}. For any n ≥ N0,∣∣∣∣∫S×X

gn(s, x)τn(d(s, x))−∫S×X

g∞(s, x)τ∞(d(s, x))

∣∣∣∣≤∣∣∣∣∫S×X

gn(s, x)τn(d(s, x))−∫S×X

g∞(s, x)τn(d(s, x))

∣∣∣∣+

∣∣∣∣∫S×X

g∞(s, x)τn(d(s, x))−∫S×X

g∞(s, x)τ∞(d(s, x))

∣∣∣∣≤∣∣∣∣∫S1×X

gn(s, x)τn(d(s, x))−∫S1×X

g∞(s, x)τn(d(s, x))

∣∣∣∣+

∣∣∣∣∣∫

(S\S1)×Xgn(s, x)τn(d(s, x))−

∫(S\S1)×X

g∞(s, x)τn(d(s, x))

∣∣∣∣∣33

+

∣∣∣∣∫S×X

g∞(s, x)τn(d(s, x))−∫S×X

g∞(s, x)τ∞(d(s, x))

∣∣∣∣≤∫E|gn(s, x)− g∞(s, x)| τn(d(s, x)) + 2 ·

∫(S\S1)×X

ψ(s)τn(d(s, x))

+

∣∣∣∣∫S×X

g∞(s, x)τn(d(s, x))−∫S×X

g∞(s, x)τ∞(d(s, x))

∣∣∣∣<ε

3+ 2 · ε

6+ε

3

= ε.

This completes the proof.

5.2 Discontinuous games with endogenous stochastic

sharing rules

Simon and Zame (1990) proved the existence of a Nash equilibrium in discontinuous

games with endogenous sharing rules. In particular, they considered a static

game with a payoff correspondence P that is bounded, nonempty, convex and

compact valued, and upper hemicontinuous. They showed that there exists a

Borel measurable selection p of the payoff correspondence, namely the endogenous

sharing rule, and a mixed strategy profile α such that α is a Nash equilibrium when

players take p as the payoff function.

In this subsection, we shall consider discontinuous games with endogenous

stochastic sharing rules. That is, we allow the payoff correspondence to depend on

some state variable in a measurable way as follows:

1. let S be a Borel subset of a Polish space, Y a Polish space, and λ a Borel

probability measure on S;

2. D is a B(S)⊗ B(Y )-measurable subset of S × Y , where D(s) is compact for

all s ∈ S and λ ({s ∈ S : D(s) 6= ∅}) > 0;

3. X =∏

1≤i≤nXi, where each Xi is a Polish space;

4. for each i, Ai is a measurable, nonempty and compact valued correspondence

from D to Xi, which is sectionally continuous on Y ;

5. A =∏

1≤i≤nAi and E = Gr(A);

6. P is a bounded, measurable, nonempty, convex and compact valued corre-

spondence from E to Rn which is essentially sectionally upper hemicontinuous

on Y ×X.

A stochastic sharing rule at (s, y) ∈ D is a Borel measurable selection of the

correspondence P (s, y, ·); i.e., a Borel measurable function p : A(s, y) → Rn such

34

that p(x) ∈ P (s, y, x) for all x ∈ A(s, y). Given (s, y) ∈ D, P (s, y, ·) represents the

set of all possible payoff profiles, and a sharing rule p is a particular choice of the

payoff profile.

Now we shall prove the following proposition.

Proposition 4. There exists a B(D)-measurable, nonempty and compact valued

correspondence Φ from D to Rn × M(X) × 4(X) such that Φ is essentially

sectionally upper hemicontinuous on Y , and for λ-almost all s ∈ S with D(s) 6= ∅and y ∈ D(s), Φ(s, y) is the set of points (v, α, µ) that

1. v =∫X p(s, y, x)α(dx) such that p(s, y, ·) is a Borel measurable selection of

P (s, y, ·);29

2. α ∈ ⊗i∈IM(Ai(s, y)) is a Nash equilibrium in the subgame (s, y) with payoff

p(s, y, ·) and action space Ai(s, y) for each player i;

3. µ = p(s, y, ·) ◦ α.30

In addition, denote the restriction of Φ on the first component Rn as Φ|Rn, which

is a correspondence from D to Rn. Then Φ|Rn is bounded, measurable, nonempty

and compact valued, and essentially sectionally upper hemicontinuous on Y .

If D is a closed subset, P is upper hemicontinuous on E and Ai is continuous

on D for each i ∈ I, then Proposition 4 is reduced to be the following lemma (see

Simon and Zame (1990) and Reny and Robson (2002, Lemma 4)).

Lemma 16. Assume that D is a closed subset, P is upper hemicontinuous on E

and Ai is continuous on D for each i ∈ I. Consider the correspondence Φ: D →Rn ×M(X)×4(X) defined as follows: (v, α, µ) ∈ Φ(s, y) if

1. v =∫X p(s, y, x)α(dx) such that p(s, y, ·) is a Borel measurable selection of

P (s, y, ·);

2. α ∈ ⊗i∈IM(Ai(s, y)) is a Nash equilibrium in the subgame (s, y) with payoff

p(s, y, ·) and action space Ai(s, y) for each player i;

3. µ = p(s, y, ·) ◦ α.

Then Φ is nonempty and compact valued, and upper hemicontinuous on D.

We shall now prove Proposition 4.

Proof of Proposition 4. There exists a Borel subset S ⊆ S with λ(S) = 1 such that

D(s) 6= ∅ for each s ∈ S, and P is sectionally upper hemicontinuous on Y when it

is restricted on D ∩ (S × Y ). Without loss of generality, we assume that S = S.

29Note that we require p(s, y, ·) to be measurable for each (s, y), but p may not be jointly measurable.30The finite measure µ = p(s, y, ·) ◦ α if µ(B) =

∫Bp(s, y, x)α(dx) for any Borel subset B ⊆ X.

35

Suppose that S is the completion of B(S) under the probability measure λ.

Let D and E be the restrictions of S ⊗ B(Y ) and S ⊗ B(Y )⊗ B(X) on D and E,

respectively.

Define a correspondence D from S to Y such that D(s) = {y ∈ Y : (s, y) ∈ D}.Then D is nonempty and compact valued. By Lemma 2 (4), D is S-measurable.

Since D(s) is compact and A(s, ·) is upper hemicontinuous for any s ∈ S, E(s)

is compact by Lemma 3 (6). Define a correspondence Γ from S to Y × X × Rn

as Γ(s) = Gr(P (s, ·, ·)). For all s, P (s, ·, ·) is bounded, upper hemicontinuous and

compact valued on E(s), hence it has a compact graph. As a result, Γ is compact

valued. By Lemma 2 (1), P has an S ⊗ B(Y ×X × Rn)-measurable graph. Since

Gr(Γ) = Gr(P ), Gr(Γ) is S⊗B(Y ×X×Rn)-measurable. Due to Lemma 2 (4), the

correspondence Γ is S-measurable. We can view Γ as a function from S into the

space K of nonempty compact subsets of Y ×X ×Rn. By Lemma 1, K is a Polish

space endowed with the Hausdorff metric topology. Then by Lemma 2 (2), Γ is

an S-measurable function from S to K. One can also define a correspondence Ai

from S to Y ×X as Ai(s) = Gr(Ai(s, ·)). It is easy to show that Ai can be viewed

as an S-measurable function from S to the space of nonempty compact subsets

of Y × X, which is endowed with the Hausdorff metric topology. By a similar

argument, D can be viewed as an S-measurable function from S to the space of

nonempty compact subsets of Y .

By Lemma 4 (Lusin’s Theorem), there exists a compact subset S1 ⊆ S such

that λ(S \ S1) < 12 , Γ, D and {Ai}1≤i≤n are continuous functions on S1. By

Lemma 2 (3), Γ, D and Ai are continuous correspondences on S1. Let D1 =

{(s, y) ∈ D : s ∈ S1, y ∈ D(s)}. Since S1 is compact and D is continuous, D1 is

compact (see Lemma 3 (6)). Similarly, E1 = E∩(S1×Y ×X) is also compact. Thus,

P is an upper hemicontinuous correspondence on E1. Define a correspondence Φ1

from D1 to Rn×M(X)×4(X) as in Lemma 16, then it is nonempty and compact

valued, and upper hemicontinuous on D1.

Following the same procedure, for any m ≥ 1, there exists a compact subset

Sm ⊆ S such that (1) Sm ∩ (∪1≤k≤m−1Sk) = ∅ and Dm = D ∩ (Sm × Y ) is

compact, (2) λ(Sm) > 0 and λ (S \ (∪1≤k≤mSm)) < 12m , and (3) there is a

nonempty and compact valued, upper hemicontinuous correspondence Φm from

Dm to Rn×M(X)×4(X), which satisfies conditions (1)-(3) in Lemma 16. Thus,

we have countably many disjoint sets {Sm}m≥1 such that (1) λ(∪m≥1Sm) = 1,

(2) Φm is nonempty and compact valued, and upper hemicontinuous on each Dm,

m ≥ 1.

Since Ai is a B(S) ⊗ B(Y )-measurable, nonempty and compact valued corre-

spondence, it has a Borel measurable selection ai by Lemma 3 (3). Fix a Borel

measurable selection p of P (such a selection exists also due to Lemma 3 (3)).

36

Define a mapping (v0, α0, µ0) from D to Rn×M(X)×4(X) such that (1) αi(s, y) =

δai(s,y) and α0(s, y) = ⊗i∈Iαi(s, y); (2) v0(s, y) = p(s, y, a1(s, y). . . . , an(s, y))

and (3) µ0(s, y) = p(s, y, ·) ◦ α0. Let D0 = D \ (∪m≥1Dm) and Φ0(s, y) =

{(v0(s, y), α0(s, y), µ0(s, y))} for (s, y) ∈ D0. Then, Φ0 is B(S)⊗B(Y )-measurable,

nonempty and compact valued.

Let Φ(s, y) = Φm(s, y) if (s, y) ∈ Dm for some m ≥ 0. Then, Φ(s, y) satisfies

conditions (1)-(3) if (s, y) ∈ Dm for m ≥ 1. That is, Φ is B(D)-measurable,

nonempty and compact valued, and essentially sectionally upper hemicontinuous

on Y , and satisfies conditions (1)-(3) for λ-almost all s ∈ S.

Then consider Φ|Rn , which is the restriction of Φ on the first component Rn.

Let Φm|Rn be the restriction of Φm on the first component Rn with the domain Dm

for each m ≥ 0. It is obvious that Φ0|Rn is measurable, nonempty and compact

valued. For each m ≥ 1, Dm is compact and Φm is upper hemicontinuous and

compact valued. By Lemma 3 (6), Gr(Φm) is compact. Thus, Gr(Φm|Rn) is also

compact. By Lemma 3 (4), Φm|Rn is measurable. In addition, Φm|Rn is nonempty

and compact valued, and upper hemicontinuous on Dm. Notice that Φ|Rn(s, y) =

Φm|Rn(s, y) if (s, y) ∈ Dm for some m ≥ 0. Thus, Φ|Rn is measurable, nonempty

and compact valued, and essentially sectionally upper hemicontinuous on Y .

The proof is complete.

5.3 Proofs of Theorem 1 and Proposition 1

5.3.1 Backward induction

For any t ≥ 1, suppose that the correspondence Qt+1 from Ht to Rn is bounded,

measurable, nonempty and compact valued, and essentially sectionally upper

hemicontinuous on Xt. For any ht−1 ∈ Ht−1 and xt ∈ At(ht−1), let

Pt(ht−1, xt) =

∫St

Qt+1(ht−1, xt, st)ft0(dst|ht−1)

=

∫St

Qt+1(ht−1, xt, st)ϕt0(ht−1, st)λt(dst).

It is obvious that the correspondence Pt is measurable and nonempty valued. Since

Qt+1 is bounded, Pt is bounded. For λt-almost all st ∈ St, Qt+1(·, st) is bounded

and upper hemicontinuous on Ht(st), and ϕt0(st, ·) is continuous on Gr(At0)(st). As

ϕt0 is integrably bounded, Pt(st−1, ·) is also upper hemicontinuous on Gr(At)(st−1)

for λt−1-almost all st−1 ∈ St−1 (see Lemma 5); that is, the correspondence Pt is

essentially sectionally upper hemicontinuous on Xt. Again by Lemma 5, Pt is

convex and compact valued since λt is an atomless probability measure. That is,

Pt : Gr(At)→ Rn is a bounded, measurable, nonempty, convex and compact valued

37

correspondence which is essentially sectionally upper hemicontinuous on Xt.

By Proposition 4, there exists a bounded, measurable, nonempty and compact

valued correspondence Φt from Ht−1 to Rn × M(Xt) × 4(Xt) such that Φt is

essentially sectionally upper hemicontinuous on Xt−1, and for λt−1-almost all

ht−1 ∈ Ht−1, (v, α, µ) ∈ Φt(ht−1) if

1. v =∫At(ht−1) pt(ht−1, x)α(dx) such that pt(ht−1, ·) is a Borel measurable

selection of Pt(ht−1, ·);

2. α ∈ ⊗i∈IM(Ati(ht−1)) is a Nash equilibrium in the subgame ht−1 with payoff

pt(ht−1, ·) and action space∏i∈I Ati(ht−1);

3. µ = pt(ht−1, ·) ◦ α.

Denote the restriction of Φt on the first component Rn as Φ(Qt+1), which is

a correspondence from Ht−1 to Rn. By Proposition 4, Φ(Qt+1) is bounded,

measurable, nonempty and compact valued, and essentially sectionally upper

hemicontinuous on Xt−1.

5.3.2 Forward induction

The following proposition presents the result on the step of forward induction.

Proposition 5. For any t ≥ 1 and any Borel measurable selection qt of Φ(Qt+1),

there exists a Borel measurable selection qt+1 of Qt+1 and a Borel measurable

mapping ft : Ht−1 → ⊗i∈IM(Xti) such that for λt−1-almost all ht−1 ∈ Ht−1,

1. ft(ht−1) ∈ ⊗i∈IM(Ati(ht−1));

2. qt(ht−1) =∫At(ht−1)

∫Stqt+1(ht−1, xt, st)ft0(dst|ht−1)ft(dxt|ht−1);

3. ft(·|ht−1) is a Nash equilibrium in the subgame ht−1 with action spaces

Ati(ht−1), i ∈ I and the payoff functions∫St

qt+1(ht−1, ·, st)ft0(dst|ht−1).

Proof. We divide the proof into three steps. In step 1, we show that there exist

Borel measurable mappings ft : Ht−1 → ⊗i∈IM(Xti) and µt : Ht−1 →4(Xt) such

that (qt, ft, µt) is a selection of Φt. In step 2, we obtain a Borel measurable selection

gt of Pt such that for λt−1-almost all ht−1 ∈ Ht−1,

1. qt(ht−1) =∫At(ht−1) gt(ht−1, x)ft(dx|ht−1);

2. ft(ht−1) is a Nash equilibrium in the subgame ht−1 with payoff gt(ht−1, ·) and

action space At(ht−1);

38

In step 3, we show that there exists a Borel measurable selection qt+1 of Qt+1 such

that for all ht−1 ∈ Ht−1 and xt ∈ At(ht−1),

gt(ht−1, xt) =

∫St

qt+1(ht−1, xt, st)ft0(dst|ht−1).

Combining Steps 1-3, the proof is complete.

Step 1. Let Ψt : Gr(Φt(Qt+1))→M(Xt)×4(Xt) be

Ψt(ht−1, v) = {(α, µ) : (v, α, µ) ∈ Φt(ht−1)}.

Recall the construction of Φt and the proof of Proposition 4, Ht−1 can be divided

into countably many Borel subsets {Hmt−1}m≥0 such that

1. Ht−1 = ∪m≥0Hmt−1 and

λt−1(∪m≥1projSt−1 (Hmt−1))

λt−1(projSt−1 (Ht−1))= 1, where projSt−1(Hm

t−1)

and projSt−1(Ht−1) are projections of Hmt−1 and Ht−1 on St−1;

2. for m ≥ 1, Hmt−1 is compact, Φt is upper hemicontinuous on Hm

t−1, and Pt is

upper hemicontinuous on

{(ht−1, xt) : ht−1 ∈ Hmt−1, xt ∈ At(ht−1)};

3. there exists a Borel measurable mapping (v0, α0, µ0) from H0t−1 to Rn ×

M(Xt) × 4(Xt) such that Φt(ht−1) ≡ {(v0(ht−1), α0(ht−1), µ0(ht−1))} for

any ht−1 ∈ H0t−1.

Denote the restriction of Φt on Hmt−1 as Φm

t . For m ≥ 1, Gr(Φmt ) is compact,

and hence the correspondence Ψmt (ht−1, v) = {(α, µ) : (v, α, µ) ∈ Φm

t (ht−1)} has

a compact graph. For m ≥ 1, Ψmt is measurable by Lemma 3 (4), and has a

Borel measurable selection ψmt due to Lemma 3 (3). Define ψ0t (ht−1, v0(ht−1)) =

(α0(ht−1), µ0(ht−1)) for ht−1 ∈ H0t−1. For (ht−1, v) ∈ Gr(Φ(Qt+1)), let ψt(ht−1, v) =

ψmt (ht−1, v) if ht−1 ∈ Hmt−1. Then ψt is a Borel measurable selection of Ψt.

Given a Borel measurable selection qt of Φ(Qt+1), let

φt(ht−1) = (qt(ht−1), ψt(ht−1, qt(ht−1))).

Then φt is a Borel measurable selection of Φt. Denote Ht−1 = ∪m≥1Hmt−1.

By the construction of Φt, there exists Borel measurable mappings ft : Ht−1 →⊗i∈IM(Xti) and µt : Ht−1 →4(Xt) such that for all ht−1 ∈ Ht−1,

1. qt(ht−1) =∫At(ht−1) pt(ht−1, x)ft(dx|ht−1) such that pt(ht−1, ·) is a Borel

measurable selection of Pt(ht−1, ·);

39

2. ft(ht−1) ∈ ⊗i∈IM(Ati(ht−1)) is a Nash equilibrium in the subgame ht−1 with

payoff pt(ht−1, ·) and action space∏i∈I Ati(ht−1);

3. µt(·|ht−1) = pt(ht−1, ·) ◦ ft(·|ht−1).

Step 2. Since Pt is upper hemicontinuous on {(ht−1, xt) : ht−1 ∈ Hmt−1, xt ∈

At(ht−1)}, due to Lemma 7, there exists a Borel measurable mapping gm such

that (1) gm(ht−1, xt) ∈ Pt(ht−1, xt) for any ht−1 ∈ Hmt−1 and xt ∈ At(ht−1), and

(2) gm(ht−1, xt) = pt(ht−1, xt) for ft(·|ht−1)-almost all xt. Fix an arbitrary Borel

measurable selection g′ of Pt. Define a Borel measurable mapping from Gr(At) to

Rn as

g(ht−1, xt) =

gm(ht−1, xt) if ht−1 ∈ Hmt−1 for m ≥ 1;

g′(ht−1, xt) otherwise.

Then g is a Borel measurable selection of Pt.

In a subgame ht−1 ∈ Ht−1, let

Bti(ht−1) = {yi ∈ Ati(ht−1) :∫At(−i)(ht−1)

gi(ht−1, yi, xt(−i))ft(−i)(dxt(−i)|ht−1) >

∫At(ht−1)

pti(ht−1, xt)ft(dxt|ht−1)}.

Since g(ht−1, xt) = pt(ht−1, xt) for ft(·|ht−1)-almost all xt,∫At(ht−1)

g(ht−1, xt)ft(dxt|ht−1) =

∫At(ht−1)

pt(ht−1, xt)ft(dxt|ht−1).

Thus, Bti is a measurable correspondence from Ht−1 to Ati(ht−1). Let Bcti(ht−1) =

Ati(ht−1) \ Bti(ht−1) for each ht−1 ∈ Ht−1. Then Bcti is a measurable and closed

valued correspondence, which has a Borel measurable graph by Lemma 2. As a

result, Bti also has a Borel measurable graph. As ft(ht−1) is a Nash equilibrium

in the subgame ht−1 ∈ Ht−1 with payoff pt(ht−1, ·), fti(Bti(ht−1)|ht−1) = 0.

Denote βi(ht−1, xt) = minPti(ht−1, xt), where Pti(ht−1, xt) is the projection

of Pt(ht−1, xt) on the i-th dimension. Then the correspondence Pti is mea-

surable and compact valued, and βi is Borel measurable. Let Λi(ht−1, xt) =

{βi(ht−1, xt)} × [0, γ]n−1, where γ > 0 is the upper bound of Pt. Denote

Λ′i(ht−1, xt) = Λi(ht−1, xt) ∩ Pt(ht−1, xt). Then Λ′i is a measurable and compact

valued correspondence, and hence has a Borel measurable selection β′i. Note that

β′i is a Borel measurable selection of Pt. Let

gt(ht−1, xt) =

40

β′i(ht−1, xt) if ht−1 ∈ Ht−1, xti ∈ Bti(ht−1) and xtj /∈ Btj(ht−1), ∀j 6= i;

g(ht−1, xt) otherwise.

Notice that

{(ht−1, xt) ∈ Gr(At) : ht−1 ∈ Ht−1, xti ∈ Bti(ht−1) and xtj /∈ Btj(ht−1),∀j 6= i; }

= Gr(At) ∩ ∪i∈I

(Gr(Bti)×∏j 6=i

Xtj) \ (∪j 6=i(Gr(Btj)×∏k 6=j

Xtk))

,

which is a Borel set. As a result, gt is a Borel measurable selection of Pt. Moreover,

gt(ht−1, xt) = pt(ht−1, xt) for all ht−1 ∈ Ht−1 and ft(·|ht−1)-almost all xt.

Fix a subgame ht−1 ∈ Ht−1. We will show that ft(·|ht−1) is a Nash equilibrium

given the payoff gt(ht−1, ·) in the subgame ht−1. Suppose that player i deviates to

some action xti.

If xti ∈ Bti(ht−1), then player i’s expected payoff is∫At(−i)(ht−1)

gti(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

=

∫∏

j 6=iBctj(ht−1)

gti(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

=

∫∏

j 6=iBctj(ht−1)

βi(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

≤∫∏

j 6=iBctj(ht−1)

pti(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

=

∫At(−i)(ht−1)

pti(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

≤∫At(ht−1)

pti(ht−1, xt)ft(dxt|ht−1)

=

∫At(ht−1)

gti(ht−1, xt)ft(dxt|ht−1).

The first and the third equalities hold since ftj(Btj(ht−1)|ht−1) = 0 for each j,

and hence ft(−i)(∏j 6=iB

ctj(ht−1)|ht−1) = ft(−i)(At(−i)(ht−1)|ht−1). The second

equality and the first inequality are due to the fact that gti(ht−1, xti, xt(−i)) =

βi(ht−1, xti, xt(−i)) = minPti(ht−1, xti, xt(−i)) ≤ pti(ht−1, xti, xt(−i)) for xt(−i) ∈∏j 6=iB

ctj(ht−1). The second inequality holds since ft(·|ht−1) is a Nash equilibrium

given the payoff pt(ht−1, ·) in the subgame ht−1. The fourth equality follows from

the fact that gt(ht−1, xt) = pt(ht−1, xt) for ft(·|ht−1)-almost all xt.

41

If xti /∈ Bti(ht−1), then player i’s expected payoff is∫At(−i)(ht−1)

gti(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

=

∫∏

j 6=iBctj(ht−1)

gti(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

=

∫∏

j 6=iBctj(ht−1)

gi(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

=

∫At(−i)(ht−1)

gi(ht−1, xti, xt(−i))ft(−i)(dxt(−i)|ht−1)

≤∫At(ht−1)

pti(ht−1, xt)ft(dxt|ht−1)

=

∫At(ht−1)

gti(ht−1, xt)ft(dxt|ht−1).

The first and the third equalities hold since

ft(−i)

∏j 6=i

Bctj(ht−1)|ht−1

= ft(−i)(At(−i)(ht−1)|ht−1).

The second equality is due to the fact that gti(ht−1, xti, xt(−i)) = gi(ht−1, xti, xt(−i))

for xt(−i) ∈∏j 6=iB

ctj(ht−1). The first inequality follows from the definition of Bti,

and the fourth equality holds since gt(ht−1, xt) = pt(ht−1, xt) for ft(·|ht−1)-almost

all xt.

Thus, player i cannot improve his payoff in the subgame ht by a unilateral

change in his strategy for any i ∈ I, which implies that ft(·|ht−1) is a Nash

equilibrium given the payoff gt(ht−1, ·) in the subgame ht−1.

Step 3. For any (ht−1, xt) ∈ Gr(At),

Pt(ht−1, xt) =

∫St

Qt+1(ht−1, xt, st)ft0(dst|ht−1).

By Lemma 6, there exists a Borel measurable mapping q from Gr(Pt) × St to Rn

such that

1. q(ht−1, xt, e, st) ∈ Qt+1(ht−1, xt, st) for any (ht−1, xt, e, st) ∈ Gr(Pt)× St;

2. e =∫Stq(ht−1, xt, e, st)ft0(dst|ht−1) for any (ht−1, xt, e) ∈ Gr(Pt), where

(ht−1, xt) ∈ Gr(At).

Let

qt+1(ht−1, xt, st) = q(ht−1, xt, gt(ht−1, xt), st)

42

for any (ht−1, xt, st) ∈ Ht. Then qt+1 is a Borel measurable selection of Qt+1.

For (ht−1, xt) ∈ Gr(At),

gt(ht−1, xt) =

∫St

q(ht−1, xt, gt(ht−1, xt), st)ft0(dst|ht−1)

=

∫St

qt+1(ht−1, xt, st)ft0(dst|ht−1).

Therefore, we have a Borel measurable selection qt+1 of Qt+1, and a Borel

measurable mapping ft : Ht−1 → ⊗i∈IM(Xti) such that for all ht−1 ∈ Ht−1,

properties (1)-(3) are satisfied. The proof is complete.

If a dynamic game has only T stages for some positive integer T ≥ 1, then let

QT+1(hT ) = {u(hT )} for any hT ∈ HT , and Qt = Φ(Qt+1) for 1 ≤ t ≤ T − 1. We

can start with the backward induction from the last period and stop at the initial

period, then run the forward induction from the initial period to the last period.

Thus, the following corollary is immediate.

Corollary 4. Any finite-horizon dynamic game with the ARM condition has a

subgame-perfect equilibrium.

5.3.3 Infinite horizon case

Pick a sequence ξ = (ξ1, ξ2, . . .) such that (1) ξm is a transition probability from

Hm−1 to M(Xm) for any m ≥ 1, and (2) ξm(Am(hm−1)|hm−1) = 1 for any m ≥ 1

and hm−1 ∈ Hm−1. Denote the set of all such ξ as Υ.

Fix any t ≥ 1, define correspondences Ξtt and ∆tt as follows: in the subgame

ht−1,

Ξtt(ht−1) =M(At(ht−1))⊗ λt,

and

∆tt(ht−1) =M(At(ht−1))⊗ ft0(ht−1).

For any m1 > t, suppose that the correspondences Ξm1−1t and ∆m1−1

t have been de-

fined. Then we can define correspondences Ξm1t : Ht−1 →M

(∏t≤m≤m1

(Xm × Sm))

and ∆m1t : Ht−1 →M

(∏t≤m≤m1

(Xm × Sm))

as follows:

Ξm1t (ht−1) ={g(ht−1) � (ξm1(ht−1, ·)⊗ λm1) :

g is a Borel measurable selection of Ξm1−1t ,

ξm1 is a Borel measurable selection of M(Am1)},

43

and

∆m1t (ht−1) ={g(ht−1) � (ξm1(ht−1, ·)⊗ fm10(ht−1, ·)) :

g is a Borel measurable selection of ∆m1−1t ,

ξm1 is a Borel measurable selection of M(Am1)},

whereM(Am1) is regarded as a correspondence from Hm1−1 to the space of Borel

probability measures onXm1 . For anym1 ≥ t, let ρm1

(ht−1,ξ)∈ Ξm1

t be the probability

measure on∏t≤m≤m1

(Xm × Sm) induced by {λm}t≤m≤m1 and {ξm}t≤m≤m1 , and

%m1

(ht−1,ξ)∈ ∆m1

t be the probability measure on∏t≤m≤m1

(Xm × Sm) induced by

{fm0}t≤m≤m1 and {ξm}t≤m≤m1 . Then, Ξm1t (ht−1) is the set of all such ρm1

(ht−1,ξ),

while ∆m1t (ht−1) is the set of all such %m1

(ht−1,ξ). Note that %m1

(ht−1,ξ)∈ ∆m1

t (ht−1) if

and only if ρm1

(ht−1,ξ)∈ Ξm1

t (ht−1). Both %m1

(ht−1,ξ)and ρm1

(ht−1,ξ)can be regarded as

probability measures on Hm1(ht−1).

Similarly, let ρ(ht−1,ξ) be the probability measure on∏m≥t(Xm×Sm) induced by

{λm}m≥t and {ξm}m≥t, and %(ht−1,ξ) the probability measure on∏m≥t(Xm × Sm)

induced by {fm0}m≥t and {ξm}m≥t. Denote the correspondence

Ξt : Ht−1 →M(∏m≥t

(Xm × Sm))

as the set of all such ρ(ht−1,ξ), and

∆t : Ht−1 →M(∏m≥t

(Xm × Sm))

as the set of all such %(ht−1,ξ).

The following lemma demonstrates the relationship between %m1

(ht−1,ξ)and

ρm1

(ht−1,ξ).

Lemma 17. For any m1 ≥ t and ht−1 ∈ Ht−1,

%m1

(ht−1,ξ)=

∏t≤m≤m1

ϕm0(ht−1, ·)

◦ ρm1

(ht−1,ξ).31

31For m ≥ t ≥ 1 and ht−1 ∈ Ht−1, the function ϕm0(ht−1, ·) is defined on Hm−1(ht−1) × Sm, whichis measurable and sectionally continuous on

∏t≤k≤m−1Xk. By Lemma 4, ϕm0(ht−1, ·) can be extended

to be a measurable function ϕm0(ht−1, ·) on the product space(∏

t≤k≤m−1Xk

)×(∏

t≤k≤m Sk

), which

is also sectionally continuous on∏t≤k≤m−1Xk. Given any ξ ∈ Υ, since ρm(ht−1,ξ)

concentrates on

Hm(ht−1), ϕm0(ht−1, ·) ◦ ρm(ht−1,ξ)= ϕm0(ht−1, ·) ◦ ρm(ht−1,ξ)

. For notational simplicity, we still use

ϕm0(ht−1, ·), instead of ϕm0(ht−1, ·), to denote the above extension. Similarly, we can work with asuitable extension of the payoff function u as needed.

44

Proof. Fix ξ ∈ Υ, and Borel subsets Cm ⊆ Xm and Dm ⊆ Sm for m ≥ t. First, we

have

%t(ht−1,ξ)(Ct ×Dt) = ξt(Ct|ht−1) · ft0(Dt|ht−1)

=

∫Xt×St

1Ct×Dt(xt, st)ϕt0(ht−1, st)(ξt(ht−1)⊗ λt)(d(xt, st)),

which implies that %t(ht−1,ξ)= ϕt0(ht−1, ·) ◦ ρt(ht−1,ξ)

.32

Suppose that %m2

(ht−1,ξ)=(∏

t≤m≤m2ϕm0(ht−1, ·)

)◦ ρm2

(ht−1,ξ)for some m2 ≥ t.

Then

%m2+1(ht−1,ξ)

∏t≤m≤m2+1

(Cm ×Dm)

= %m2

(ht−1,ξ)� (ξm2+1(ht−1, ·)⊗ f(m2+1)0(ht−1, ·))

∏t≤m≤m2+1

(Cm ×Dm)

=

∫∏

t≤m≤m2(Xm×Sm)

∫Xm2+1×Sm2+1

1∏t≤m≤m2+1(Cm×Dm)(xt, . . . , xm2+1, st, . . . , sm2+1)·

ξm2+1 ⊗ f(m2+1)0(d(xm2+1, sm2+1)|ht−1, xt, . . . , xm2 , st, . . . , sm2)

%m2

(ht−1,ξ)(d(xt, . . . , xm2 , st, . . . , sm2)|ht−1)

=

∫∏

t≤m≤m2(Xm×Sm)

∫Sm2+1

∫Xm2+1

1∏t≤m≤m2+1(Cm×Dm)(xt, . . . , xm2+1, st, . . . , sm2+1)·

ϕ(m2+1)0(ht−1, xt, . . . , xm2 , st, . . . , sm2+1)ξm2+1(dxm2+1|ht−1, xt, . . . , xm2 , st, . . . , sm2)

λ(m2+1)0(dsm2+1)∏

t≤m≤m2

ϕm0(ht−1, xt, . . . , xm−1, st, . . . , sm)

ρm2

(ht−1,ξ)(d(xt, . . . , xm2 , st, . . . , sm2)|ht−1)

=

∫∏

t≤m≤m2+1(Xm×Sm)1∏

t≤m≤m2+1(Cm×Dm)(xt, . . . , xm2+1, st, . . . , sm2+1)·∏t≤m≤m2+1

ϕm0(ht−1, xt, . . . , xm−1, st, . . . , sm)ρm2+1(ht−1,ξ)

(d(xt, . . . , xm2 , st, . . . , sm2)|ht−1),

which implies that

%m2+1(ht−1,ξ)

=

∏t≤m≤m2+1

ϕm0(ht−1, ·)

◦ ρm2+1(ht−1,ξ)

.

The proof is thus complete.

The next lemma shows that the correspondences ∆m1t and ∆t are nonempty

32For a set A in a space X, 1A is the indicator function of A, which is one on A and zero on X \A.

45

and compact valued, and sectionally continuous.

Lemma 18. 1. For any t ≥ 1, the correspondence ∆m1t is nonempty and

compact valued, and sectionally continuous on Xt−1 for any m1 ≥ t.

2. For any t ≥ 1, the correspondence ∆t is nonempty and compact valued, and

sectionally continuous on Xt−1.

Proof. (1) We first show that the correspondence Ξm1t is nonempty and compact

valued, and sectionally continuous on Xt−1 for any m1 ≥ t.Consider the case m1 = t ≥ 1, where

Ξtt(ht−1) =M(At(ht−1))⊗ λt.

Since Ati is nonempty and compact valued, and sectionally continuous on Xt−1,

Ξtt is nonempty and compact valued, and sectionally continuous on Xt−1.

Now suppose that Ξm2t is nonempty and compact valued, and sectionally

continuous on Xt−1 for some m2 ≥ t ≥ 1. Notice that

Ξm2+1t (ht−1) ={g(ht−1) � (ξm2+1(ht−1, ·)⊗ λ(m2+1)) :

g is a Borel measurable selection of Ξm2t ,

ξm2+1 is a Borel measurable selection of M(Am2+1)}.

First, we claim that Ht(s0, s1, . . . , st) is compact for any (s0, s1, . . . , st) ∈ St.We prove this claim by induction.

1. Notice that H0(s0) = X0 for any s0 ∈ S0, which is compact.

2. Suppose that Hm′(s0, s1, . . . , sm′) is compact for some 0 ≤ m′ ≤ t − 1 and

any (s0, s1, . . . , sm′) ∈ Sm′.

3. Since Am′+1(·, s0, s1, . . . , sm′) is continuous and compact valued, it has a

compact graph by Lemma 3 (6), which is Hm′+1(s0, s1, . . . , sm′+1) for any

(s0, s1, . . . , sm′+1) ∈ Sm′+1.

Thus, we prove the claim.

Define a correspondence Att from Ht−1 × St to Xt as Att(ht−1, st) = At(ht−1).

Then Att is nonempty and compact valued, sectionally continuous on Xt−1, and

has a B(Xt × St)-measurable graph. Since the graph of Att(·, s0, s1, . . . , st)

is Ht(s0, s1, . . . , st) and Ht(s0, s1, . . . , st) is compact, Att(·, s0, s1, . . . , st) has a

compact graph. For any ht−1 ∈ Ht−1 and τ ∈ Ξtt(ht−1), the marginal of τ on

St is λt and τ(Gr(Att(ht−1, ·))) = 1.

46

For any m1 > t, suppose that the correspondence

Am1−1t : Ht−1 ×

∏t≤m≤m1−1

Sm →∏

t≤m≤m1−1

Xm

has been defined such that

1. it is nonempty and compact valued, sectionally upper hemicontinuous on

Xt−1, and has a B(Xm1−1 × Sm1−1)-measurable graph;

2. for any (s0, s1, . . . .sm1−1), Am1−1t (·, s0, s1, . . . .sm1−1) has a compact graph;

3. for any ht−1 ∈ Ht−1 and τ ∈ Ξm1−1t (ht−1), the marginal of τ on∏

t≤m≤m1−1 Sm is ⊗t≤m≤m1−1λm and τ(Gr(Am1−1t (ht−1, ·))) = 1.

We define a correspondence Am1t : Ht−1 ×

∏t≤m≤m1

Sm →∏t≤m≤m1

Xm as

follows:

Am1t (ht−1, st, . . . , sm1) ={(xt, . . . , xm1) :

xm1 ∈ Am1(ht−1, xt, . . . , xm1−1, st, . . . , sm1−1),

(xt, . . . , xm1−1) ∈ Am1−1t (ht−1, st, . . . , sm1−1)}.

It is obvious that Am1t is nonempty valued. For any (s0, s1, . . . , sm1), since

Am1−1t (·, s0, s1, . . . .sm1−1) has a compact graph and Am1(·, s0, s1, . . . , sm1−1) is

continuous and compact valued, Am1t (·, s0, s1, . . . .sm1) has a compact graph by

Lemma 3 (6), which implies that Am1t is compact valued and sectionally upper

hemicontinuous on Xt−1. In addition, Gr(Am1t ) = Gr(Am1) × Sm1 , which is

B(Xm1 × Sm1)-measurable. For any ht−1 ∈ Ht−1 and τ ∈ Ξm1t (ht−1), it is obvious

that the marginal of τ on∏t≤m≤m1

Sm is ⊗t≤m≤m1λm and τ(Gr(Am1t (ht−1, ·))) =

1.

By Lemma 14, Ξm2+1t is nonempty and compact valued, and sectionally

continuous on Xt−1.

Now we show that the correspondence ∆m1t is nonempty and compact valued,

and sectionally continuous on Xt−1 for any m1 ≥ t.Given st−1 and a sequence {xk0, xk1, . . . , xkt−1} ∈ Ht−1(st−1) for 1 ≤ k ≤ ∞. Let

hkt−1 = (st−1, (xk0, xk1, . . . , x

kt−1)). It is obvious that ∆m1

t is nonempty valued, we

first show that ∆m1t is sectionally upper hemicontinuous on Xt−1. Suppose that

%m1

(hkt−1,ξk)∈ ∆m1

t (hkt−1) for 1 ≤ k <∞ and (xk0, xk1, . . . , x

kt−1)→ (x∞0 , x

∞1 , . . . , x

∞t−1),

we need to show that there exists some ξ∞ such that a subsequence of %m1

(hkt−1,ξk)

weakly converges to %m1

(h∞t−1,ξ∞) and %m1

(h∞t−1,ξ∞) ∈ ∆m1

t (h∞t−1).

Since Ξm1t is sectionally upper hemicontinuous on Xt−1, there exists some ξ∞

such that a subsequence of ρm1

(hkt−1,ξk)

, say itself, weakly converges to ρm1

(h∞t−1,ξ∞) and

47

ρm1

(h∞t−1,ξ∞) ∈ Ξm1

t (h∞t−1). Then %m1

(h∞t−1,ξ∞) ∈ ∆m1

t (h∞t−1).

For any bounded continuous function ψ on∏t≤m≤m1

(Xm × Sm), let

χk(xt, . . . , xm1 , st, . . . , sm1) =

ψ(xt, . . . , xm1 , st, . . . , sm1) ·∏

t≤m≤m1

ϕm0(hkt−1, xt, . . . , xm−1, st, . . . , sm).

Then {χk} is a sequence of functions satisfying the following three properties.

1. For each k, χk is jointly measurable and sectionally continuous on∏t≤m≤m1

Xm.

2. For any (st, . . . , sm1) and any sequence (xkt , . . . , xkm1

) → (x∞t , . . . , x∞m1

) in∏t≤m≤m1

Xm, χk(xkt , . . . , x

km1, st, . . . , sm1) → χ∞(x∞t , . . . , x

∞m1, st, . . . , sm1)

as k →∞.

3. The sequence {χk}1≤k≤∞ is integrably bounded in the sense that there exists

a function χ′ :∏t≤m≤m1

Sm → R+ such that χ′ is ⊗t≤m≤m1λm-integrable

and for any k and (xt, . . . , xm1 , st, . . . , sm1), χk(xt, . . . , xm1 , st, . . . , sm1) ≤χ′(st, . . . , sm1).

By Lemma 15, as k →∞,∫∏

t≤m≤m1(Xm×Sm)

χk(xt, . . . , xm1 , st, . . . , sm1)ρm1

(hkt−1,ξk)

(d(xt, . . . , xm1 , st, . . . , sm1))

→∫∏

t≤m≤m1(Xm×Sm)

χ∞(xt, . . . , xm1 , st, . . . , sm1)ρm1

(h∞t−1,ξ∞)(d(xt, . . . , xm1 , st, . . . , sm1)).

Then by Lemma 17,∫∏

t≤m≤m1(Xm×Sm)

ψ(xt, . . . , xm1 , st, . . . , sm1)%m1

(hkt−1,ξk)

(d(xt, . . . , xm1 , st, . . . , sm1))

→∫∏

t≤m≤m1(Xm×Sm)

ψ(xt, . . . , xm1 , st, . . . , sm1)%m1

(h∞t−1,ξ∞)(d(xt, . . . , xm1 , st, . . . , sm1)),

which implies that %m1

(hkt−1,ξk)

weakly converges to %m1

(h∞t−1,ξ∞). Therefore, ∆m1

t is

sectionally upper hemicontinuous on Xt−1. If one chooses h1t−1 = h2

t−1 = · · · =

h∞t−1, then we indeed show that ∆m1t is compact valued.

In the argument above, we indeed proved that if ρm1

(hkt−1,ξk)

weakly converges to

ρm1

(h∞t−1,ξ∞), then %m1

(hkt−1,ξk)

weakly converges to %m1

(h∞t−1,ξ∞).

The left is to show that ∆m1t is sectionally lower hemicontinuous on Xt−1.

Suppose that (xk0, xk1, . . . , x

kt−1)→ (x∞0 , x

∞1 , . . . , x

∞t−1) and %m1

(h∞t−1,ξ∞) ∈ ∆m1

t (h∞t−1),

we need to show that there exists a subsequence {(xkm0 , xkm1 , . . . , xkmt−1)} of

{(xk0, xk1, . . . , xkt−1)} and %m1

(hkmt−1,ξkm )∈ ∆m1

t (hkmt−1) for each km such that %m1

(hkmt−1,ξkm )

48

weakly converges to %m1

(h∞t−1,ξ∞).

Since %m1

(h∞t−1,ξ∞) ∈ ∆m1

t (h∞t−1), we have ρm1

(h∞t−1,ξ∞) ∈ Ξm1

t (h∞t−1). Because

Ξm1t is sectionally lower hemicontinuous on Xt−1, there exists a subsequence of

{(xk0, xk1, . . . , xkt−1)}, say itself, and ρm1

(hkt−1,ξk)∈ Ξm1

t (hkt−1) for each k such that

ρm1

(hkt−1,ξk)

weakly converges to ρm1

(h∞t−1,ξ∞). As a result, %m1

(hkt−1,ξk)

weakly converges to

%m1

(h∞t−1,ξ∞), which implies that ∆m1

t is sectionally lower hemicontinuous on Xt−1.

Therefore, ∆m1t is nonempty and compact valued, and sectionally continuous

on Xt−1 for any m1 ≥ t.

(2) We show that ∆t is nonempty and compact valued, and sectionally

continuous on Xt−1.

It is obvious that ∆t is nonempty valued, we first prove that it is compact

valued.

Given ht−1 and a sequence {τk} ⊆ ∆t(ht−1), there exists a sequence of {ξk}k≥1

such that ξk = (ξk1 , ξk2 , . . .) ∈ Υ and τk = %(ht−1,ξk) for each k.

By (1), Ξtt is compact. Then there exists a measurable mapping gt such that

(1) gt = (ξ11 , . . . , ξ

1t−1, gt, ξ

1t+1, . . .) ∈ Υ, and (2) a subsequence of {ρt

(ht−1,ξk)},

say {ρt(ht−1,ξ

k1l )}l≥1, which weakly converges to ρt(ht−1,gt)

. Note that {ξkt+1} is

a Borel measurable selection of M(At+1). By Lemma 14, there is a Borel

measurable selection gt+1 of M(At+1) such that there is a subsequence of

{ρt+1(ht−1,ξ

k1l )}l≥1, say {ρt+1

(ht−1,ξk2l )}l≥1, which weakly converges to ρt+1

(ht−1,gt+1), where

gt+1 = (ξ11 , . . . , ξ

1t−1, gt, gt+1, ξ

1t+2, . . .) ∈ Υ.

Repeat this procedure, one can construct a Borel measurable mapping g such

that ρ(ht−1,ξk11 ), ρ(ht−1,ξk22 ), ρ(ht−1,ξk33 ), . . . weakly converges to ρ(ht−1,g). That is,

ρ(ht−1,g) is a convergent point of {ρ(ht−1,ξk)}, which implies that %(ht−1,g) is a

convergent point of {%(ht−1,ξk)}.The sectional upper hemicontinuity of ∆t follows a similar argument as above.

In particular, given st−1 and a sequence {xk0, xk1, . . . , xkt−1} ⊆ Ht−1(st−1) for

k ≥ 0. Let hkt−1 = (st−1, (xk0, xk1, . . . , x

kt−1)). Suppose that (xk0, x

k1, . . . , x

kt−1) →

(x00, x

01, . . . , x

0t−1). If {τk} ⊆ ∆t(h

kt−1) for k ≥ 1 and τk → τ0, then one can show

that τ0 ∈ ∆t(h0t−1) by repeating a similar argument as in the proof above.

Finally, we consider the sectional lower hemicontinuity of ∆t. Suppose that

τ0 ∈ ∆t(h0t−1). Then there exists some ξ ∈ Υ such that τ0 = %(h0t−1,ξ)

. Denote

τm = %m(h0t−1,ξ)

∈ ∆mt (h0

t−1) for m ≥ t. As ∆mt is continuous, for each m, there

exists some ξm ∈ Υ such that d(%m(hkmt−1,ξ

m), τm) ≤ 1

m for km sufficiently large, where

d is the Prokhorov metric. Let τm = %(hkmt−1,ξ

m). Then τm weakly converges to τ0,

which implies that ∆t is sectionally lower hemicontinuous.

49

Define a correspondence Qτt : Ht−1 → Rn++ as follows:

Qτt (ht−1) ={∫∏

m≥t(Xm×Sm) u(ht−1, x, s)%(ht−1,ξ)(d(x, s)) : %(ht−1,ξ) ∈ ∆t(ht−1)}; t > τ ;

Φ(Qτt+1)(ht−1) t ≤ τ.

The lemma below presents several properties of the correspondence Qτt .

Lemma 19. For any t, τ ≥ 1, Qτt is bounded, measurable, nonempty and compact

valued, and essentially sectionally upper hemicontinuous on Xt−1.

Proof. We prove the lemma in three steps.

Step 1. Fix t > τ . We will show that Qτt is bounded, nonempty and compact

valued, and sectionally upper hemicontinuous on Xt−1.

The boundedness and nonemptiness of Qτt are obvious. We shall prove that

Qτt is sectionally upper hemicontinuous on Xt−1. Given st−1 and a sequence

{xk0, xk1, . . . , xkt−1} ⊆ Ht−1(st−1) for k ≥ 0. Let hkt−1 = (st−1, (xk0, xk1, . . . , x

kt−1)).

Suppose that ak ∈ Qτt (hkt−1) for k ≥ 1, (xk0, xk1, . . . , x

kt−1) → (x0

0, x01, . . . , x

0t−1) and

ak → a0, we need to show that a0 ∈ Qτt (h0t−1).

By the definition, there exists a sequence {ξk}k≥1 such that

ak =

∫∏

m≥t(Xm×Sm)u(hkt−1, x, s)%(hkt−1,ξ

k)(d(x, s)),

where ξk = (ξk1 , ξk2 , . . .) ∈ Υ for each k. As ∆t is compact valued and sectionally

continuous on Xt−1, there exist some %(h0t−1,ξ0) ∈ ∆t(h

0t−1) and a subsequence of

%(hkt−1,ξk), say itself, which weakly converges to %(h0t−1,ξ

0) for ξ0 = (ξ01 , ξ

02 , . . .) ∈ Υ.

We shall show that

a0 =

∫∏

m≥t(Xm×Sm)u(h0

t−1, x, s)%(h0t−1,ξ0)(d(x, s)).

For this aim, we only need to show that for any δ > 0,∣∣∣∣∣a0 −∫∏

m≥t(Xm×Sm)u(h0

t−1, x, s)%(h0t−1,ξ0)(d(x, s))

∣∣∣∣∣ < δ. (3)

Since the game is continuous at infinity, there exists a positive integer M ≥ t

such that wm < 15δ for any m > M .

For each j > M , by Lemma 4, there exists a measurable selection ξ′j ofM(Aj)

such that ξ′j is sectionally continuous on Xj−1. Let µ : HM →∏m>M (Xm ×

Sm) be the transition probability which is induced by (ξ′M+1

, ξ′M+2

, . . .) and

50

{f(M+1)0, f(M+2)0, . . .}. By Lemma 10, µ is measurable and sectionally continuous

on XM . Let

VM (ht−1, xt, . . . , xM , st, . . . , sM ) =∫∏

m>M (Xm×Sm)u(ht−1, xt, . . . , xM , st, . . . , sM , x, s) dµ(x, s|ht−1, xt, . . . , xM , st, . . . , sM ).

Then VM is bounded and measurable. In addition, VM is sectionally continuous

on XM by Lemma 15.

For any k ≥ 0, we have

∣∣ ∫∏m≥t(Xm×Sm)

u(hkt−1, x, s)%(hkt−1,ξk)(d(x, s))

−∫∏

t≤m≤M (Xm×Sm)VM (hkt−1, xt, . . . , xM , st, . . . , sM )%M

(hkt−1,ξk)

(d(xt, . . . , xM , st, . . . , sM ))∣∣

≤ wM+1

<1

5δ.

Since %(hkt−1,ξk) weakly converges to %(h0t−1,ξ

0) and %M(hkt−1,ξ

k)is the marginal of

%(hkt−1,ξk) on

∏t≤m≤M (Xm×Sm) for any k ≥ 0, the sequence %M

(hkt−1,ξk)

also weakly

converges to %M(h0t−1,ξ

0). By Lemma 15, we have

|∫∏

t≤m≤M (Xm×Sm)VM (hkt−1, xt, . . . , xM , st, . . . , sM )%M

(hkt−1,ξk)

(d(xt, . . . , xM , st, . . . , sM ))

−∫∏

t≤m≤M (Xm×Sm)VM (h0

t−1, xt, . . . , xM , st, . . . , sM )%M(h0t−1,ξ0)(d(xt, . . . , xM , st, . . . , sM ))|

<1

5δ

for k ≥ K1, where K1 is a sufficiently large positive integer. In addition, there

exists a positive integer K2 such that |ak − a0| < 15δ for k ≥ K2.

Fix k > max{K1,K2}. Combining the inequalities above, we have∣∣∣∣∣∫∏

m≥t(Xm×Sm)u(h0

t−1, x, s)%(h0t−1,ξ0)(d(x, s))− a0

∣∣∣∣∣≤∣∣ ∫∏

m≥t(Xm×Sm)u(h0

t−1, x, s)%(h0t−1,ξ0)(d(x, s))

−∫∏

t≤m≤M (Xm×Sm)VM (h0

t−1, xt, . . . , xM , st, . . . , sM )%M(h0t−1,ξ0)(d(xt, . . . , xM , st, . . . , sM ))

∣∣+∣∣ ∫∏

t≤m≤M (Xm×Sm)VM (h0

t−1, xt, . . . , xM , st, . . . , sM )%M(h0t−1,ξ0)(d(xt, . . . , xM , st, . . . , sM ))

51

−∫∏

t≤m≤M (Xm×Sm)VM (hkt−1, xt, . . . , xM , st, . . . , sM )%M

(hkt−1,ξk)

(d(xt, . . . , xM , st, . . . , sM ))∣∣

+∣∣ ∫∏

t≤m≤M (Xm×Sm)VM (hkt−1, xt, . . . , xM , st, . . . , sM )%M

(hkt−1,ξk)

(d(xt, . . . , xM , st, . . . , sM ))

−∫∏

m≥t(Xm×Sm)u(hkt−1, x, s)%(hkt−1,ξ

k)(d(x, s))∣∣

+∣∣ ∫∏

m≥t(Xm×Sm)u(hkt−1, x, s)%(hkt−1,ξ

k)(d(x, s))− a0∣∣

< δ.

Thus, we proved inequality (3), which implies that Qτt is sectionally upper

hemicontinuous on Xt−1 for t > τ .

Furthermore, to prove that Qτt is compact valued, we only need to consider

the case that {xk0, xk1, . . . , xkt−1} = {x00, x

01, . . . , x

0t−1} for any k ≥ 0, and repeat the

above proof.

Step 2. Fix t > τ , we will show that Qτt is measurable.

Fix a sequence (ξ′1, ξ′2, . . .), where ξ′j is a selection ofM(Aj) measurable in sj−1

and continuous in xj−1 for each j. For any M ≥ t, let

WMM (ht−1, xt, . . . , xM , st, . . . , sM ) ={∫

∏m>M (Xm×Sm)

u(ht−1, xt, . . . , xM , st, . . . , sM , x, s)%(ht−1,xt,...,xM ,st,...,sM ,ξ′)(d(x, s))

}.

By Lemma 10, %(ht−1,xt,...,xM ,st,...,sM ,ξ′) is measurable fromHM toM(∏

m>M (Xm × Sm)),

and sectionally continuous on XM . Thus, WMM is bounded, measurable, nonempty,

convex and compact valued. By Lemma 15, WMM is sectionally continuous on XM .

Suppose that for some t ≤ j ≤M , W jM has been defined such that it is bounded,

measurable, nonempty, convex and compact valued, and sectionally continuous on

Xj . Let

W j−1M (ht−1, xt, . . . , xj−1, st, . . . , sj−1) ={∫

Xj×Sj

wjM (ht−1, xt, . . . , xj , st, . . . , sj)%j(ht−1,xt,...,xj−1,st,...,sj−1,ξ)

(d(xj , sj)) :

%j(ht−1,xt,...,xj−1,st,...,sj−1,ξ)∈ ∆j

j(ht−1, xt, . . . , xj−1, st, . . . , sj−1),

wjM is a Borel measurable selection of W jM

}.

52

Let Sj = Sj .33 Since∫

Xj×Sj

W jM (ht−1, xt, . . . , xj , st, . . . , sj)%

j(ht−1,xt,...,xj−1,st,...,sj−1,ξ)

(d(xj , sj))

=

∫Sj

∫Xj×Sj

W jM (ht−1, xt, . . . , xj , st, . . . , sj)ρ

j(ht−1,xt,...,xj−1,st,...,sj−1,ξ)

(d(xj , sj))

· ϕj0(ht−1, xt, . . . , xj−1, st, . . . , sj)λj(dsj),

we have

W j−1M (ht−1, xt, . . . , xj−1, st, . . . , sj−1) ={∫

Sj

∫Xj×Sj

wjM (ht−1, xt, . . . , xj , st, . . . , sj)ρj(ht−1,xt,...,xj−1,st,...,sj−1,ξ)

(d(xj , sj))

· ϕj0(ht−1, xt, . . . , xj−1, st, . . . , sj)λj(dsj) :

ρj(ht−1,xt,...,xj−1,st,...,sj−1,ξ)∈ Ξjj(ht−1, xt, . . . , xj−1, st, . . . , sj−1),

wjM is a Borel measurable selection of W jM

}.

Let

W jM (ht−1, xt, . . . , xj−1, st, . . . , sj) ={∫

Xj×Sj

wjM (ht−1, xt, . . . , xj , st, . . . , sj) · ρj(ht−1,xt,...,xj−1,st,...,sj−1,ξ)(d(xj , sj)) :

ρj(ht−1,xt,...,xj−1,st,...,sj−1,ξ)∈ Ξjj(ht−1, xt, . . . , xj−1, st, . . . , sj−1),

wjM is a Borel measurable selection of W jM

}.

Since W jM (ht−1, xt, . . . , xj , st, . . . , sj) is continuous in xj and does not depend on

sj , it is continuous in (xj , sj). In addition, W jM is bounded, measurable, nonempty,

convex and compact valued. By Lemma 11, W jM is bounded, measurable,

nonempty and compact valued, and sectionally continuous on Xj−1.

It is easy to see that

W j−1M (ht−1, xt, . . . , xj−1, st, . . . , sj−1) =∫

Sj

W jM (ht−1, xt, . . . , xj−1, st, . . . , sj)ϕj0(ht−1, xt, . . . , xj−1, st, . . . , sj)λj(dsj).

By Lemma 5, it is bounded, measurable, nonempty and compact valued, and

sectionally continuous on Xj−1. By induction, one can show that W t−1M is bounded,

33We will need to use Lemma 11 below, which requires the continuity of the correspondences in termsof the integrated variables. Since W j

M is only measurable, but not continuous, in sj , we add a dummy

variable sj so that W jM is trivially continuous in such a variable.

53

measurable, nonempty and compact valued, and sectionally continuous on Xt−1.

Let W t−1 = ∪M≥tW t−1M . That is, W t−1 is the closure of ∪M≥tW t−1

M , which is

measurable due to Lemma 3.

First, W t−1 ⊆ Qτt because W t−1M ⊆ Qτt for each M ≥ t and Qτt is compact

valued. Second, fix ht−1 and q ∈ Qτt (ht−1). Then there exists a mapping ξ ∈ Υ

such that

q =

∫∏

m≥t(Xm×Sm)u(ht−1, x, s)%(ht−1,ξ)(d(x, s)).

For M ≥ t, let

VM (ht−1, xt, . . . , xM , st, . . . , sM ) =∫∏

m>M (Xm×Sm)u(ht−1, xt, . . . , xM , st, . . . , sM , x, s)%(ht−1,xt,...,xM ,st,...,sM ,ξ)(x, s)

and

qM =

∫∏

t≤m≤M (Xm×Sm)VM (ht−1, x, s)%

M(ht−1,ξ)

(d(x, s)).

Hence, qM ∈ W t−1M . Because the dynamic game is continuous at infinity, qM → q,

which implies that q ∈W t−1(ht−1) and Qτt ⊆W t−1.

Therefore, W t−1 = Qτt , and hence Qτt is measurable for t > τ .

Step 3. For t ≤ τ , we can start with Qττ+1. Repeating the backward induction

in Subsection 5.3.1, we have that Qτt is also bounded, measurable, nonempty and

compact valued, and essentially sectionally upper hemicontinuous on Xt−1.

Denote

Q∞t =

Qt−1t , if ∩τ≥1 Q

τt = ∅;

∩τ≥1Qτt , otherwise.

The following three lemmas show that Q∞t (ht−1) = Φ(Q∞t+1)(ht−1) = Et(ht−1) for

λt−1-almost all ht−1 ∈ Ht−1.34

Lemma 20. 1. The correspondence Q∞t is bounded, measurable, nonempty and

compact valued, and essentially sectionally upper hemicontinuous on Xt−1.

2. For any t ≥ 1, Q∞t (ht−1) = Φ(Q∞t+1)(ht−1) for λt−1-almost all ht−1 ∈ Ht−1.

Proof. (1) It is obvious that Q∞t is bounded. By the definition of Qτt , for λt−1-

almost all ht−1 ∈ Ht−1, Qτ1t (ht−1) ⊆ Qτ2t (ht−1) for τ1 ≥ τ2. Since Qτt is nonempty

and compact valued, Q∞t = ∩τ≥1Qτt is nonempty and compact valued for λt−1-

almost all ht−1 ∈ Ht−1. If ∩τ≥1Qτt = ∅, then Q∞t = Qt−1

t . Thus, Q∞t (ht−1) is

34The proofs for Lemmas 20 and 22 follow the standard ideas with various modifications; see, forexample, Harris (1990), Harris, Reny and Robson (1995) and Mariotti (2000).

54

nonempty and compact valued for all ht−1 ∈ Ht−1. By Lemma 3 (2), ∩τ≥1Qτt is

measurable, which implies that Q∞t is measurable.

Fix any st−1 ∈ St−1 such that Qτt (·, st−1) is upper hemicontinuous on

Ht−1(st−1) for any τ . By Lemma 3 (7), Qτt (·, st−1) has a closed graph for each τ ,

which implies that Q∞t (·, st−1) has a closed graph. Referring to Lemma 3 (7) again,

Q∞t (·, st−1) is upper hemicontinuous on Ht−1(st−1). Since Qτt is essentially upper

hemicontinuous on Xt−1 for each τ , Q∞t is essentially upper upper hemicontinuous

on Xt−1.

(2) For any τ ≥ 1 and λt−1-almost all ht−1 ∈ Ht−1, Φ(Q∞t+1)(ht−1) ⊆Φ(Qτt+1)(ht−1) ⊆ Qτt (ht−1), and hence Φ(Q∞t+1)(ht−1) ⊆ Q∞t (ht−1).

The space {1, 2, . . .∞} is a countable compact set endowed with the following

metric: d(k,m) = | 1k −1m | for any 1 ≤ k,m ≤ ∞. The sequence {Qτt+1}1≤τ≤∞

can be regarded as a correspondence Qt+1 from Ht × {1, 2, . . . ,∞} to Rn, which

is measurable, nonempty and compact valued, and essentially sectionally upper

hemicontinuous on Xt×{1, 2, . . . ,∞}. The backward induction in Subsection 5.3.1

shows that Φ(Qt+1) is measurable, nonempty and compact valued, and essentially

sectionally upper hemicontinuous on Xt × {1, 2, . . . ,∞}.Since Φ(Qt+1) is essentially sectionally upper hemicontinuous onXt×{1, 2, . . . ,∞},

there exists a measurable subset St−1 ⊆ St−1 such that λt−1(St−1) = 1, and

Φ(Qt+1)(·, ·, st−1) is upper hemicontinuous for any st−1 ∈ St−1. Fix st−1 ∈St−1. For ht−1 = (xt−1, st−1) ∈ Ht−1 and a ∈ Q∞t (ht−1), by its definition,

a ∈ Qτt (ht−1) = Φ(Qτt+1)(ht−1) for τ ≥ t. Thus, a ∈ Φ(Q∞t+1)(ht−1).

In summary, Q∞t (ht−1) = Φ(Q∞t+1)(ht−1) for λt−1-almost all ht−1 ∈ Ht−1.

Though the definition of Qτt involves correlated strategies for τ < t, the

following lemma shows that one can work with mixed strategies in terms of

equilibrium payoffs via the combination of backward and forward inductions in

multiple steps.

Lemma 21. If ct is a measurable selection of Φ(Q∞t+1), then ct(ht−1) is a subgame-

perfect equilibrium payoff vector for λt−1-almost all ht−1 ∈ Ht−1.

Proof. Without loss of generality, we only prove the case t = 1.

Suppose that c1 is a measurable selection of Φ(Q∞2 ). Apply Proposition 5

recursively to obtain Borel measurable mappings {fki}i∈I for k ≥ 1. That is,

for any k ≥ 1, there exists a Borel measurable selection ck of Q∞k such that for

λk−1-almost all hk−1 ∈ Hk−1,

1. fk(hk−1) is a Nash equilibrium in the subgame hk−1, where the action space

55

is Aki(hk−1) for player i ∈ I, and the payoff function is given by∫Sk

ck+1(hk−1, ·, sk)fk0(dsk|hk−1).

2.

ck(hk−1) =

∫Ak(hk−1)

∫Sk

ck+1(hk−1, xk, sk)fk0(dsk|hk−1)fk(dxk|hk−1).

We need to show that c1(h0) is a subgame-perfect equilibrium payoff vector for

λ0-almost all h0 ∈ H0.

Step 1. We show that for any k ≥ 1 and λk−1-almost all hk−1 ∈ Hk−1,

ck(hk−1) =

∫∏

m≥k(Xm×Sm)u(hk−1, x, s)%(hk−1,f)(d(x, s)).

Since the game is continuous at infinity, there exists some positive integer M >

k such that wM is sufficiently small. By Lemma 20, ck(hk−1) ∈ Q∞k (hk−1) =

∩τ≥1Qτk(hk−1) for λk−1-almost all hk−1 ∈ Hk−1. Since Qτk = Φτ−k+1(Qττ+1) for

k ≤ τ , ck(hk−1) ∈ ∩τ≥kΦτ−k+1(Qττ+1)(hk−1) ⊆ ΦM−k+1(QMM+1)(hk−1) for λk−1-

almost all hk−1 ∈ Hk−1. Thus, there exists a Borel measurable selection w of

QMM+1 and some ξ ∈ Υ such that for λM−1-almost all hM−1 ∈ HM−1,

i. fM (hM−1) is a Nash equilibrium in the subgame hM−1, where the action

space is AMi(hM−1) for player i ∈ I, and the payoff function is given by∫SM

w(hM−1, ·, sM )fM0(dsM |hM−1);

ii.

cM (hM−1) =

∫AM (hM−1)

∫SM

w(hM−1, xM , sM )fM0(dsM |hM−1)fM (dxM |hM−1);

iii. w(hM ) =∫∏

m≥M+1(Xm×Sm) u(hM , x, s)%(hM ,ξ)(d(x, s)).

Then for λk−1-almost all hk−1 ∈ Hk−1,

ck(hk−1) =

∫∏

m≥k(Xm×Sm)u(hk−1, x, s)%(hk−1,fM )(d(x, s)),

where fMk is fk if k ≤ M , and ξk if k ≥ M + 1. Since the game is continuous at

56

infinity, ∫∏

m≥k(Xm×Sm)u(hk−1, x, s)%(hk−1,fM )(d(x, s))

converges to ∫∏

m≥k(Xm×Sm)u(hk−1, x, s)%(hk−1,f)(d(x, s))

when M goes to infinity. Thus, for λk−1-almost all hk−1 ∈ Hk−1,

ck(ht−1) =

∫∏

m≥k(Xm×Sm)u(hk−1, x, s)%(hk−1,f)(d(x, s)). (4)

Step 2. Below, we show that {fki}i∈I is a subgame-perfect equilibrium.

Fix a player i and a strategy gi = {gki}k≥1. For each k ≥ 1, define a new

strategy fki as follows: fki = (g1i, . . . , gki, f(k+1)i, f(k+2)i, . . .). That is, we simply

replace the initial k stages of fi by gi. Denote fk = (fki , f−i).

Fix k ≥ 1 and a measurable subset Dk ⊆ Sk such that (1) and (2) of step 1

and Equation (4) hold for all sk ∈ Dk and xk ∈ Hk(sk), and λk(Dk) = 1. For each

M > k, by the Fubini property, there exists a measurable subset EMk ⊆ Sk such

that λk(EMk ) = 1 and ⊗k+1≤j≤Mλj(DM (sk)) = 1 for all sk ∈ EMk , where

DM (sk) = {(sk+1, . . . , sM ) : (sk, sk+1, . . . , sM ) ∈ DM}.

Let Dk = (∩M>kEMk ) ∩Dk. Then λk(Dk) = 1.

For any hk = (xk, sk) such that sk ∈ Dk and xk ∈ Hk(sk), we have∫

∏m≥k+1(Xm×Sm)

u(hk, x, s)%(hk,f)(d(x, s))

=

∫Ak+1(hk)

∫Sk+1

c(k+2)i(hk, xk+1, sk+1)f(k+1)0(dsk+1|hk)fk+1(dxk+1|hk)

≥∫Ak+1(hk)

∫Sk+1

c(k+2)i(hk, xk+1, sk+1)f(k+1)0(dsk+1|hk)(f(k+1)(−i) ⊗ g(k+1)i

)(dxk+1|hk)

=

∫Ak+1(hk)

∫Sk+1

∫Ak+2(hk,xk+1,sk+1)

∫Sk+2

c(k+3)i(hk, xk+1, sk+1, xk+2, sk+2)

f(k+2)0(dsk+2|hk, xk+1, sk+1)f(k+2)(−i) ⊗ f(k+2)i(dxk+2|hk, xk+1, sk+1)

f(k+1)0(dsk+1|hk)f(k+1)(−i) ⊗ g(k+1)i(dxk+1|hk)

≥∫Ak+1(hk)

∫Sk+1

∫Ak+2(hk,xk+1,sk+1)

∫Sk+2

c(k+3)i(hk, xk+1, sk+1, xk+2, sk+2)

f(k+2)0(dsk+2|hk, xk+1, sk+1)f(k+2)(−i) ⊗ g(k+2)i(dxk+2|hk, xk+1, sk+1)

f(k+1)0(dsk+1|hk)f(k+1)(−i) ⊗ g(k+1)i(dxk+1|hk)

57

=

∫∏

m≥k+1(Xm×Sm)u(hk, x, s)%(hk,fk+2)(d(x, s)).

The first and the last equalities follow from Equation (4) in the end of step 1. The

second equality is due to (2) in step 1. The first inequality is based on (1) in step 1.

The second inequality holds by the following arguments:

i. by the choice of hk and (1) in step 1, for λk+1-almost all sk+1 ∈ Sk+1 and all

xk+1 ∈ Xk+1 such that (hk, xk+1, sk+1) ∈ Hk+1, we have∫Ak+2(hk,xk+1,sk+1)

∫Sk+2

c(k+3)i(hk, xk+1, sk+1, xk+2, sk+2)

f(k+2)0(dsk+2|hk, xk+1, sk+1)f(k+2)(−i) ⊗ f(k+2)i(dxk+2|hk, xk+1, sk+1)

≥∫Ak+2(hk,xk+1,sk+1)

∫Sk+2

c(k+3)i(hk, xk+1, sk+1, xk+2, sk+2)

f(k+2)0(dsk+2|hk, xk+1, sk+1)f(k+2)(−i) ⊗ g(k+2)i(dxk+2|hk, xk+1, sk+1);

ii. since f(k+1)0 is absolutely continuous with respect to λk+1, the above

inequality also holds for f(k+1)0(hk)-almost all sk+1 ∈ Sk+1 and all xk+1 ∈Xk+1 such that (hk, xk+1, sk+1) ∈ Hk+1.

Repeating the above argument, one can show that∫∏

m≥k+1(Xm×Sm)u(hk, x, s)%(hk,f)(d(x, s))

≥∫∏

m≥k+1(Xm×Sm)u(hk, x, s)%(hk,fM+1)

(d(x, s))

for any M > k. Since∫∏

m≥k+1(Xm×Sm)u(hk, x, s)%(hk,fM+1)

(d(x, s))

converges to ∫∏

m≥k+1(Xm×Sm)u(hk, x, s)%(hk,(gi,f−i))(d(x, s))

as M goes to infinity, we have∫∏

m≥k+1(Xm×Sm)u(hk, x, s)%(hk,f)(d(x, s))

≥∫∏

m≥k+1(Xm×Sm)u(hk, x, s)%(hk,(gi,f−i))(d(x, s)).

Therefore, {fki}i∈I is a subgame-perfect equilibrium.

58

By Lemma 20 and Proposition 4, the correspondence Φ(Q∞t+1) is measurable,

nonempty and compact valued. By Lemma 3 (3), it has a measurable selection.

Then Theorem 1 follows from the above lemma.

For t ≥ 1 and ht−1 ∈ Ht−1, recall that Et(ht−1) is the set of payoff vectors of

subgame-perfect equilibria in the subgame ht−1. The following lemma shows that

Et(ht−1) is essentially the same as Q∞t (ht−1).

Lemma 22. For any t ≥ 1, Et(ht−1) = Q∞t (ht−1) for λt−1-almost all ht−1 ∈ Ht−1.

Proof. (1) We will first prove the following claim: for any t and τ , if Et+1(ht) ⊆Qτt+1(ht) for λt-almost all ht ∈ Ht, then Et(ht−1) ⊆ Qτt (ht−1) for λt−1-almost all

ht−1 ∈ Ht−1. We only need to consider the case that t ≤ τ .

By the construction of Φ(Qτt+1) in Subsection 5.3.1, there exists a measurable

subset St−1 ⊆ St−1 with λt−1(St−1) = 1 such that for any ct and ht−1 =

(xt−1, st−1) ∈ Ht−1 with st−1 ∈ St−1, if

1. ct =∫At(ht−1)

∫Stqt+1(ht−1, xt, st)ft0(dst|ht−1)α(dxt), where qt+1(ht−1, ·) is

measurable and qt+1(ht−1, xt, st) ∈ Qτt+1(ht−1, xt, st) for λt-almost all st ∈ Stand xt ∈ At(ht−1);

2. α ∈ ⊗i∈IM(Ati(ht−1)) is a Nash equilibrium in the subgame ht−1 with payoff∫Stqt+1(ht−1, ·, st)ft0(dst|ht−1) and action space

∏i∈I Ati(ht−1),

then ct ∈ Φ(Qτt+1)(ht−1).

Fix a subgame ht−1 = (xt−1, st−1) such that st−1 ∈ St−1. Pick a point

ct ∈ Et(st−1). There exists a strategy profile f such that f is a subgame-perfect

equilibrium in the subgame ht−1 and the payoff is ct. Let ct+1(ht−1, xt, st) be the

payoff vector induced by {fti}i∈I in the subgame (ht, xt, st) ∈ Gr(At) × St. Then

we have

1. ct =∫At(ht−1)

∫Stct+1(ht−1, xt, st)ft0(dst|ht−1)ft(dxt|ht−1);

2. ft(·|ht−1) is a Nash equilibrium in the subgame ht−1 with action space

At(ht−1) and payoff∫Stct+1(ht−1, ·, st)ft0(dst|ht−1).

Since f is a subgame-perfect equilibrium in the subgame ht−1, ct+1(ht−1, xt, st) ∈Et+1(ht−1, xt, st) ⊆ Qτt+1(ht−1, xt, st) for λt-almost all st ∈ St and xt ∈ At(ht−1),

which implies that ct ∈ Φ(Qτt+1)(ht−1) = Qτt (ht−1).

Therefore, Et(ht−1) ⊆ Qτt (ht−1) for λt−1-almost all ht−1 ∈ Ht−1.

(2) For any t > τ , Et ⊆ Qτt . If t ≤ τ , we can start with Eτ+1 ⊆ Qττ+1 and repeat

the argument in (1), then we can show that Et(ht−1) ⊆ Qτt (ht−1) for λt−1-almost

all ht−1 ∈ Ht−1. Thus, Et(ht−1) ⊆ Q∞t (ht−1) for λt−1-almost all ht−1 ∈ Ht−1.

(3) Suppose that ct is a measurable selection from Φ(Q∞t+1). Apply Proposi-

tion 5 recursively to obtain Borel measurable mappings {fki}i∈I for k ≥ t. By

59

Lemma 21, ct(ht−1) is a subgame-perfect equilibrium payoff vector for λt−1-almost

all ht−1 ∈ Ht−1. Consequently, Φ(Q∞t+1)(ht−1) ⊆ Et(ht−1) for λt−1-almost all

ht−1 ∈ Ht−1.

By Lemma 20, Et(ht−1) = Q∞t (ht−1) = Φ(Q∞t+1)(ht−1) for λt−1-almost all

ht−1 ∈ Ht−1.

Therefore, we have proved Proposition 1.

5.4 Proof of Proposition 2

We will highlight the needed changes in comparison with the proofs presented in

Subsections 5.3.1-5.3.3.

1. Backward induction. We first consider stage t with Nt = 1.

If Nt = 1, then St = {st}. Thus, Pt(ht−1, xt) = Qt+1(ht−1, xt, st), which is

nonempty and compact valued, and essentially sectionally upper hemicontinuous

on Xt × St−1. Notice that Pt may not be convex valued.

We first assume that Pt is upper hemicontinuous. Suppose that j is the player

who is active in this period. Consider the correspondence Φt : Ht−1 → Rn ×M(Xt)×4(Xt) defined as follows: (v, α, µ) ∈ Φt(ht−1) if

1. v = pt(ht−1, At(−j)(ht−1), x∗tj) such that pt(ht−1, ·) is a measurable selection

of Pt(ht−1, ·);35

2. x∗tj ∈ Atj(ht−1) is a maximization point of player j given the payoff function

ptj(ht−1, At(−j)(ht−1), ·) and the action space Atj(ht−1), αi = δAti(ht−1) for

i 6= j and αj = δx∗tj ;

3. µ = δpt(ht−1,At(−j)(ht−1),x∗tj).

This is a single agent problem. We need to show that Φt is nonempty and compact

valued, and upper hemicontinuous.

If Pt is nonempty, convex and compact valued, and upper hemicontinuous,

then we can use Lemma 16, the main result of Simon and Zame (1990), to prove

the nonemptiness, compactness, and upper hemicontinuity of Φt. In Simon and

Zame (1990), the only step they need the convexity of Pt for the proof of their main

theorem is Lemma 2 therein. However, the one-player pure-strategy version of their

Lemma 2, stated in the following, directly follows from the upper hemicontinuity

of Pt without requiring the convexity.

Let Z be a compact metric space, and {zn}n≥0 ⊆ Z. Let P : Z → R+ be

a bounded, upper hemicontinuous correspondence with nonempty and

compact values. For each n ≥ 1, let qn be a Borel measurable selection

35Note that At(−j) is point valued since all players other than j are inactive.

60

of P such that qn(zn) = dn. If zn converges to z0 and dn converges to

some d0, then d0 ∈ P (z0).

Repeat the argument in the proof of the main theorem of Simon and Zame

(1990), one can show that Φt is nonempty and compact valued, and upper

hemicontinuous.

Then we go back to the case that Pt is nonempty and compact valued,

and essentially sectionally upper hemicontinuous on Xt × St−1. Recall that we

proved Proposition 4 based on Lemma 16. If Pt is essentially sectionally upper

hemicontinuous on Xt× St−1, we can show the following result based on a similar

argument as in Subsections 5.2: there exists a bounded, measurable, nonempty

and compact valued correspondence Φt from Ht−1 to Rn ×M(Xt) ×4(Xt) such

that Φt is essentially sectionally upper hemicontinuous on Xt−1 × St−1, and for

λt−1-almost all ht−1 ∈ Ht−1, (v, α, µ) ∈ Φt(ht−1) if

1. v = pt(ht−1, At(−j)(ht−1), x∗tj) such that pt(ht−1, ·) is a measurable selection

of Pt(ht−1, ·);

2. x∗tj ∈ Atj(ht−1) is a maximization point of player j given the payoff function

ptj(ht−1, At(−j)(ht−1), ·) and the action space Atj(ht−1), αi = δAti(ht−1) for

i 6= j and αj = δx∗tj ;

3. µ = δpt(ht−1,At(−j)(ht−1),x∗tj).

Next we consider the case that Nt = 0. Suppose that the correspondence

Qt+1 from Ht to Rn is bounded, measurable, nonempty and compact valued, and

essentially sectionally upper hemicontinuous on Xt × St. For any (ht−1, xt, st) ∈Gr(At), let

Rt(ht−1, xt, st) =

∫St

Qt+1(ht−1, xt, st, st)ft0(dst|ht−1, xt, st)

=

∫St

Qt+1(ht−1, xt, st, st)ϕt0(ht−1, xt, st, st)λt(dst).

Then following the same argument as in Subsection 5.3.1, one can show that

Rt is a nonempty, convex and compact valued, and essentially sectionally upper

hemicontinuous correspondence on Xt × St.For any ht−1 ∈ Ht−1 and xt ∈ At(ht−1), let

Pt(ht−1, xt) =

∫At0(ht−1,xt)

Rt(ht−1, xt, st)ft0(dst|ht−1, xt).

By Lemma 8, Pt is nonempty, convex and compact valued, and essentially

sectionally upper hemicontinuous on Xt × St−1. The rest of the step remains

the same as in Subsection 5.3.1.

61

2. Forward induction: unchanged.

3. Infinite horizon: we need to slightly modify the definition of Ξm1t for any

m1 ≥ t ≥ 1. Fix any t ≥ 1. Define a correspondence Ξtt as follows: in the subgame

ht−1,

Ξtt(ht−1) = (M(At(ht−1)) � ft0(ht−1, ·))⊗ λt.

For any m1 > t, suppose that the correspondence Ξm1−1t has been defined. Then we

can define a correspondence Ξm1t : Ht−1 →M

(∏t≤m≤m1

(Xm × Sm))

as follows:

Ξm1t (ht−1) =

{g(ht−1) �

((ξm1(ht−1, ·) � fm10(ht−1, ·))⊗ λm1

):

g is a Borel measurable selection of Ξm1−1t ,

ξm1 is a Borel measurable selection of M(Am1)}.

Then the result in Subsection 5.3.3 is true with the above Ξm1t .

Consequently, a subgame-perfect equilibrium exists.

5.5 Proof of Proposition 3

We will describe the necessary changes in comparison with the proofs presented in

Subsections 5.3.1-5.3.3 and 5.4.

1. Backward induction. For any t ≥ 1, suppose that the correspondence

Qt+1 from Ht to Rn is bounded, nonempty and compact valued, and upper

hemicontinuous on Xt.

If Nt = 1, then St = {st}. Thus, Pt(ht−1, xt) = Qt+1(ht−1, xt, st), which

is nonempty and compact valued, and upper hemicontinuous. Then define the

correspondence Φt from Ht−1 to Rn ×M(Xt)×4(Xt) as (v, α, µ) ∈ Φt(ht−1) if

1. v = pt(ht−1, At(−j)(ht−1), x∗tj) such that pt(ht−1, ·) is a measurable selection

of Pt(ht−1, ·);

2. x∗tj ∈ Atj(ht−1) is a maximization point of player j given the payoff function

ptj(ht−1, At(−j)(ht−1), ·) and the action space Atj(ht−1), αi = δAti(ht−1) for

i 6= j and αj = δx∗tj ;

3. µ = δpt(ht−1,At(−j)(ht−1),x∗tj).

As discussed in Subsection 5.4, Φt is nonempty and compact valued, and upper

hemicontinuous.

62

When Nt = 0, for any ht−1 ∈ Ht−1 and xt ∈ At(ht−1),

Pt(ht−1, xt) =

∫At0(ht−1,xt)

Qt+1(ht−1, xt, st)ft0(dst|ht−1, xt).

Let coQt+1(ht−1, xt, st) be the convex hull of Qt+1(ht−1, xt, st). Because Qt+1 is

bounded, nonempty and compact valued, coQt+1 is bounded, nonempty, convex

and compact valued. By Lemma 3 (8), coQt+1 is upper hemicontinuous.

Notice that ft0(·|ht−1, xt) is atomless and Qt+1 is nonempty and compact

valued. By Lemma 5,

Pt(ht−1, xt) =

∫At0(ht−1,xt)

coQt+1(ht−1, xt, st)ft0(dst|ht−1, xt).

By Lemma 8, Pt is bounded, nonempty, convex and compact valued, and upper

hemicontinuous. Then instead of relying on Proposition 4, we now use Lemma 16

to conclude that Φt is bounded, nonempty and compact valued, and upper

hemicontinuous.

2. Forward induction. The first step is much simpler.

For any {(hkt−1, vk)}1≤k≤∞ ⊆ Gr(Φt(Qt+1)) such that (hkt−1, v

k) converges to

(h∞t−1, v∞), pick (αk, µk) such that (vk, αk, µk) ∈ Φt(h

kt−1) for 1 ≤ k < ∞. Since

Φt is upper hemicontinuous and compact valued, there exists a subsequence of

(vk, αk, µk), say itself, such that (vk, αk, µk) converges to some (v∞, α∞, µ∞) ∈Φt(h

∞t−1) due to Lemma 2 (5). Thus, (α∞, µ∞) ∈ Ψt(h

∞t−1, v

∞), which implies that

Ψt is also upper hemicontinuous and compact valued. By Lemma 3 (3), Ψt has a

Borel measurable selection ψt. Given a Borel measurable selection qt of Φ(Qt+1),

one can let φt(ht−1) = (qt(ht−1), ψt(ht−1, qt(ht−1))). Then φt is a Borel measurable

selection of Φt.

Steps 2 and 3 are unchanged.

3. Infinite horizon. We do not need to consider Ξm1t for any m1 ≥ t ≥ 1.

Instead of relying on Lemma 14, we can use Lemma 10 (3) to prove Lemma 18. The

proof of Lemma 19 is much simpler. Notice that the boundedness, nonemptiness,

compactness and upper hemicontinuity of Qτt for the case t > τ is immediate. Then

one can apply the backward induction as in Lemma 19 to show the corresponding

properties of Qτt for the case t ≤ τ . Following the same arguments, one can show

that Lemmas 20-22 now hold for all ht−1 ∈ Ht−1 and all t ≥ 1.

63

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